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Journal: 2008-01-04

Ah, Wikipedia. What a way to lose time.

So this evening I was wandering wide-eyed through the wilds of the Wikipedia mathematics section, and I stumbled upon the page for .999... which of course made fascinating reading. I remember that I too did not believe it the first time someone told me that .999... = 1. The intersting thing about the Wikipedia page is it gives examples of reasons why people don't believe it. I fell in to the 4th category:

  • Some students regard 0.999… as having a fixed value which is less than 1 by an infinitely small amount. (i.e. 1 - 0.999\ldots = 10^{-\infty})
    • The proof I used to convince myself was more or less along the lines of the proof given in Wikipedia using Cauchy sequences. (I didn't know I was so mathematically advanced!)
    Truncations of the decimal number b0.b1b2b3… generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number. Thus in this formalism the task is to show that the sequence of rational numbers

    \left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right) = \left(1, {1 \over 10}, {1 \over 100}, \dots \right)

    has the limit 0. Considering the nth term of the sequence, for n=0,1,2,…, it must therefore be shown that

    \lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.

    This limit is plain.

    OK, so really, I just considered the last limit (though I don't think I even got that formal, I think I pictured it as 0.000...1 → 0.) So 1-.999...=0, so 1=.999... .

    The other proof I used was to consider approaching a point on a number line. Approaching from the left, we get, .999, .9999, .99999, . . . etc. Approaching from the right, we get 1.001, 1.0001, 1.0001, . . . etc. So .999... (with an infinite number of 9's) is just as good as 1.000... (with an infinite number of 0's) for representing the number this limit is approaching, except that traditionally we leave the 000... implied.

    An interesting fact that Wikipedia pointed out that I hadn't thought of is that the only numbers that don't have two decimal representations are irrational numbers! (That is, rational numbers have two representations, irrationals have one.)

    For some real coolness, look down the page at the p-adic numbers! See where they say …999 = −1 for 10-adic numbers? That's 2's-complement (10's-complement?) arithmetic - the way computers usually represent negative numbers! So "p-adic numbers" must be the mathematical formalism for this style of math. It's not just slight of hand, but there's some number-theoretic depth to it. How cool is that? Definitely worth investigating!

    ˜ ™

    And all this because I was trying to figure out if volatility was dPrice/dTime. (Well, I know it's not, but what is it, units-analysis-wise?)

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    Louis K. Thomas <louiNØSP@Msth@hotmÑOSP@Mail.coNÕSP@Mm> Fight Spam! Auth 2008-01-05 (197 days ago)