[quote=mdk]
Another way to approach it is, 1 - 0.9999999..... = X. In this case, X must equal 0.0000.......1, where the 1 does not occur until after an inifinitely long sequence of zeroes. It's a null infinity, which can never be expressed; thus, there is no difference between 1 and 0.999... Where there is no difference, the values are equivalent.
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[quote=Halo]
If the "9" digit in "0.999..." is recurring, then considering 1 - 0.999... = x, there is naught x can be but 0 (implying that 1 = 0.999...). There is no real number that exists between 0.999... and 1, no space between them, and thus they must be one and the same. There is no "0.000...0001" left over when you make the subtraction, because those recurring nines continue for eternity. If the difference between two numbers is zero, they are the same.
[/quote]

;D
Though, unlike you, I completely fuckin' forgot to clearly clarify that the reason there's no "0.00..001" is because that requires there to be a finite number of nines, which there isn't. Whoops. ^^;;