How not to describe the Cantor set

The latest book from my unread-books shelf is Fractals, Chaos, Power Laws, by Manfred Schroeder, which I am currently perusing over lunch. It belonged to my father, so it’s been sitting on my shelf waiting for me to read it for 15+ years.

I already knew that it had pretty fractal pictures in it, and that that was probably most of the appeal of the book for me. But I figured I might as well try reading the text as well.

Unfortunately, I’m finding the text a little annoying. In particular, when confronted with a choice about how to say something, Schroeder tends to choose cuteness over clarity. I’m having to read various bits multiple times in order to understand what he’s getting at, because he leaves out important stuff that would clarify what he means.

But I hadn’t noticed anything that looked outright wrong to me until I got to the description of Cantor dust, on p. 16.

In the midst of that description, Schroeder says:

[…] the arithmetic description of the Cantor set: its members are precisely all those fractions in the interval [0,1] that eschew the digit 1, such as 0.2 or 0.2022.

And I looked at the accompanying diagram, and I thought, That can’t be right. For example, 0.5 is obviously not in the Cantor set, but it also obviously doesn’t include the digit 1.

I re-read that line a few times, and looked at the diagram a few times, and thought about what I previously knew about the Cantor set, but Schroeder’s claim just didn’t make any sense to me. So I finally did a web search to find out more. And it turns out that Schroeder’s description is in fact completely accurate—

—in base 3.

He was so busy telling readers about this interesting numerical fact that he neglected to mention what base he was using.

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