Another argument, more or less following similar thinking, is whether a number could be found between 0.999... and 1.000... (like the mean).
If no such number can be found, then we might reasonably say they're one and the same. — jorndoe
This argument isn’t actually valid, because it could arguably be the case (if not for other, valid proofs that 0.999... = 1) that 0.999... is the very last number before 1, so there is nothing between them and even though they’re not (this hypothetical person would argue) the exact same. — Pfhorrest
for some systems 0.111 would be close enough and for NASA (they have less room for error), 0.1111111111111111111111111111111111111111111 might be required — christian2017
The point I’m making is that that “...” is an important difference.
0.111 does not equal 1/9. It’s close, and you’re saying it might be close enough for some purposes, but for others you might need more 1s. But so long as you have finitely many 1s, it won’t equal exactly 1/9.
But 0.111... (with that “...”, that’s very important) equals EXACTLY 1/9, by definition. It has infinitely many 1s. That’s what the “...” means: “keep repeating the preceding pattern forever.”
0.111 x 9 = 0.999, which is not 1.
0.111... x 9 = 1/9 x 9 = 0.999... = 1, exactly. — Pfhorrest
The naturals aren't densely ordered like the rationals and the reals. — jorndoe
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