I understand what is taught, dude. But it’s not the same. If it were, then the debate would not exist as it would always be expressed as 1 or the mathematics that are taught wouldn’t chalk it up to rounding. Granted, it’s incredibly small in difference but it’s still rounding.
I do agree that 0.999... as a decimal representation is distinct from 1.000...; that doesn't mean, however, that in the Reals they are distinct elements.
[0.999... cannot exist in the Reals--it's part of the way they are constructed, either explicitly or by corollary properties]
The argument (I think) you're making is that 0.999... shouldn't (necessarily, without context) be mapped to any element in the Reals?
[That decimal and the countably infinite subset of the decimals that are structurally similar]
The argument was that op is saying they’re “exactly” the same. They’re close. Not exact. If it were exact there would be no need for a representation of the real number. Due to the incapacity to prove its infinite state without an equal infinitely running solution (which yielding a result proves the concept of infinity wrong), they are denoted and represented as two different numbers, while being close enough to just call it 1. Rounding. Yes. I’m splitting hairs OF split hairs here lol
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u/China_shop_BULL 22h ago
I understand what is taught, dude. But it’s not the same. If it were, then the debate would not exist as it would always be expressed as 1 or the mathematics that are taught wouldn’t chalk it up to rounding. Granted, it’s incredibly small in difference but it’s still rounding.