Hardy and Wright, Chapter 9

Continuing on, we talked about “The Representation of Numbers by Decimals” this week. I thought the first few sections were fun, in the precision and care used to prove things like “Rational numbers have repeating decimals, and vice versa”.

Chris and I both, apparently, took a minute to digest the example that 29310478561 is divisible by 7, at the end of section 9.5. I feel like in my first year of undergrad I learned about this, or perhaps a similar, test for division by 7. I thought that instead of taking the sum of digits (like you would for mod 3 or mod 9), or alternating sum (for mod 11), you could take some sort of weighted sum of the digits, where the weights depended on 7^n\pmod{10} for various n. Looking around online while writing this post, it seems I was (mis-)remembering the method of Pascal, listed on this page at Mathworld. This Mathworld page mentions several other tests, which look interesting. My search turned up another as well, over at God Plays Dice.

Eric mentioned that he’s read a lot about the game of Nim (section 9.8) because Conway writes about it in On Numbers and Games, and Eric likes surreal numbers.

I like the section on “Integers with missing digits” (section 9.9), because I like to show my calculus students that the sum of the reciprocals of such numbers is a convergent series (even though, writing out the first several terms, it looks like you haven’t thrown out many terms from the harmonic series). This is known as the Kempner Series, which I first learned about in the book Gamma, by Havil.

We concluded with a little discussion on normal numbers, the last few sections of the chapter. It seems we all ran out of steam while reading this chapter, so we didn’t get through the proofs in these sections. But the ideas are interesting, and the results are fun. According to Wikipedia, there is a conjecture that all irrational algebraic numbers are normal, even though there is no known proof that any particular irrational algebraic number is normal. I remain a little confused about the definition of normal, actually. “We say that x is normal in the scale of r if all of the numbers x,rx,r^2x,\ldots are simply normal in all of the scales r,r^2,r^3,\ldots“. I think the idea with multiplying by these powers and changing the scale is that you are looking at longer and longer digit sequences, instead of just single digits (which would be simply normal). I was a little unclear about why you need to multiply x by all of those powers of r, but I guess if you don’t then you won’t ever get any digits (in the scale r^n) greater than r^{n-1}, perhaps?

Update 20090515: I completely forgot to mention another thing we talked about, and I had promised the group I’d link to. Section 9.9, on “Integers with missing digits” begins with the line “There is a familiar paradox” and a footnote that reads “Relevant in controversies about telephone directories”. I wasn’t exactly sure what this controvery was, but we decided probably it was the fact that the probability of picking a random number out of the telephone book, and having it not contain a 9 (say) is fairly small. At first, I had thought maybe the controversy the footnote was hinting at might be related to Benford’s Law, which I also remembered was just recently in the news (slashdot).

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