I didn't like the conclusion I was coming to, which was that math on a real number with a necessarily infinite decimal is effectively impossible.
Well, you don't have to do math on the decimal. You can opt to say, for example, that 1/3, √(5), π, e, etc. don't have exact decimal representations, and have to be represented by other means.
Apropos of nothing: are you familiar with Hilbert's hotel (http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel)? I know most people know Zeno's paradoxes, but that's another good infinity-screws-with-your-head one. (And actually, the cigar example they gave suggests another trick for thinking of the infinite nines: you can conjure a carry-at-infinity the same way they conjured the cigars, I suspect.)
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Well, you don't have to do math on the decimal. You can opt to say, for example, that 1/3, √(5), π, e, etc. don't have exact decimal representations, and have to be represented by other means.
Apropos of nothing: are you familiar with Hilbert's hotel (http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel)? I know most people know Zeno's paradoxes, but that's another good infinity-screws-with-your-head one. (And actually, the cigar example they gave suggests another trick for thinking of the infinite nines: you can conjure a carry-at-infinity the same way they conjured the cigars, I suspect.)