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Friday, February 02, 2007

Happy numbers, unhappy families

One of my students came across the Wikipedia article on Happy Numbers and asked about it in precalculus class. This is the sort of topic for which Wikipedia is an excellent source; in fact, if I wanted to know about happy numbers, it’s the first place to which I turned. So we looked at it in class. Ordinarily I don’t like to project a webpage in front of a class if I haven’t vetted it in advance, but this particular topic seemed (and was) harmless. Anyway, check out the article yourself: it’s easy reading and provides some amusing mathematics along the way.

As we were going through the article, and I was helping the students tie it to some of the ideas we have been exploring, I observed that we often talk about families of numbers (primes, squares, and so forth). I pointed out, however, that number families are the reverse of human families, since all unhappy numbers are alike but all happy numbers are unhappy in their own way. Not a single student recognized the allusion, so we had to digress into a two-minute literature lesson. Sigh.

By the way, you should read the Wikipedia article to see why it’s true that all unhappy numbers are alike but all happy numbers are unhappy in their own way.

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