Ah, I knew this would this would invite the curiosity of students and the ire of pedants!

@Aaron: Great question. These types of proofs make assumptions about how addition and subtraction would work with these infinite decimals (does 9 * 0.999 = 8.999…?), but they do work for the regular number system (see http://math.fau.edu/Richman/HTML/999.htm).

@some bla guy: I’d love to have a word with Cauchy – I bet he’d be interested in learning about new number systems that can rigorously approach the same problems differently!

@mau: Neat – I’ll have to see how well Google translate does at math :).

@ram: Whoops, thanks for the correction! Yes, I meant anti-matter :).

@Chad: Great points, thanks for the discussion! I guess it depends on the meaning of 0.999…, which is indeed ambiguous. I think the better phrasing may be “The hyperreal number uH=0.999…;…999000… with H-infinitely many 9s, for some infinite hyperinteger H, satisfies a strict inequality uH < 1” (from Wikipedia).

I think the higher meta-point is figuring out the question behind the question – the layman isn’t asking about 0.999… as constructed in the real number system. They want to know what happens when one number gets “infinitely close” to another – can this be represented? 0.999… is the most convenient form of this question (also see 1/infinity – does this equal 0? Yes, if you take the limit approach, no if you take the hyperreal).

@haileris: Does 1/3 = .333… exactly, or is 0.333… different at the infinitely small level?

@Adam: Great point – because our current number system cannot represent the difference between 0.999… and 1 (there’s no number in-between), in our current system they are equal. However, other systems allow it, so I take the approach of “it depends”.

@Jeff: Thanks for dropping in, but I disagree that it needs to be put to rest. Transform the problem: if it’s 1600 and someone asks 1600s Jeff what does sqrt(-1) mean, what do you say? That they are asking this question in the context of the real numbers, and the answer is undefined? How else do you answer it?

The alternative is to explore a new number system (complex numbers, hyperreal numbers) and see if it has interesting properties. You can’t take the question at face value, it’s really about exploring the nature of infinitely small numbers.