Listening to the Rationals and Irrationals

Mar 19, 2008 by

Earlier this year I shared an application with you that would play a song based on the digits of the decimal expansion of pi. After I suggested that playing rational and irrational numbers could be a nice learning activity, one of the readers discovered a Wolfram Demonstration that also plays the irrational numbers called “Math Songs.”

I’m pleased to report that Michael Croucher (from the blog Walking Randomly) has obliged us with the requisite companion Wolfram Demonstration that plays the RATIONAL numbers (any rational number with a numerator between 1 and 1000 and denominator between 1 and 1000). His demonstration is called “Music from the Rationals.”

Now that we have both, what can we do with them? As you are, no doubt, aware, many students have great difficulty with the distinction between the rational and irrational numbers. Now you can play each number song-style and ask the students to identify whether the song ends, has a repeating element, or contains a random pattern.

I was playing several fractions with denominators of 7, and then some with denominators of 13, when I made an interesting observation (well, maybe an obvious one). The same elements of sound patterns repeat, even as the numerators change. With so much calculator use today, I wonder if students realize all the patterns in the decimal expansions. Many students have probably never done a long division problem that resulted in a repeating decimal involving more than two digits. I can’t wait to use these audio applications with the students from Math for Elementary Teachers next time I teach it.

Two days ago, we covered the discriminant in my algebra class. That would have been the perfect time to talk about rationals and irrationals again. Well – there’s always next semester!

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  1. saparrynow

    What an amazing applet. I love what you are doing for mathematics education.

  2. Anonymous

    Any irony to the fact that there was a ’60s MI rock band called The Rationals?

  3. GS

    I teach high school mathematics. To get the idea of recurring vs non-recurring decimal expansions across, I show them a few HTML files I created. These have the decimal expansions of 1/7, PI, SQRT(2), etc. with the decimal digits colour-coded. It’s easy to see that the colour sequences repeat in some cases but not in others.

    Communicating the same thing using music sounds like a great thing to show the students. Cheers!

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