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.
I think what I'm settling on is that any rational number with an infinitely repeating decimal value can be exactly represented using the ... notation, but it doesn't make sense to do any math on that representation. Irrational numbers don't have exact decimal representations at all.
Apropos of nothing: are you familiar with Hilbert's hotel?
I wasn't! That was cool. It didn't really mess with my head at all -- it's been a while, but I did once read George Gamow's "1 2 3 ... Infinity" (http://www.amazon.com/One-Two-Three-Infinity-Speculations/dp/0486256642), and it does a great job of explaining countable infinities.
But, then they got to the cigars bit, and of course I wrinkled my nose.
(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.)
no subject
I think what I'm settling on is that any rational number with an infinitely repeating decimal value can be exactly represented using the ... notation, but it doesn't make sense to do any math on that representation. Irrational numbers don't have exact decimal representations at all.
Apropos of nothing: are you familiar with Hilbert's hotel?
I wasn't! That was cool. It didn't really mess with my head at all -- it's been a while, but I did once read George Gamow's "1 2 3 ... Infinity" (http://www.amazon.com/One-Two-Three-Infinity-Speculations/dp/0486256642), and it does a great job of explaining countable infinities.
But, then they got to the cigars bit, and of course I wrinkled my nose.
(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.)
Oh, eww, yuk, god, no. Please.