@sed good luck.

From his dump before this one, it's apparent he conceives of 0.(3) as "(3)/1(0)". But I don't think he means this notation to literally denote transfinite numbers because his mathematical understanding is not sophisticated enough for that; instead judging by his wording "for any given number of digits we can rewrite 1(0) as (9)+1", he doesn't have a well-defined understanding of recurring decimals at all; to him they are, I think, a way of talking about decimals with an *undefined* rather than *infinite* number of decimal places. So he talks a lot like he thinks 0.(9) is 0. followed by an unspecified number of nines.

You're unlikely to get any confirmation of this; he seems to be aware this isn't how everyone else treats the notation so won't sign up to it (and he's blocked me so I won't get an answer; I'm still here because I find it all so fascinating).

But from that point of view, it *does* make a kind of sense that you "can't do" 0.(9) × 10 because the former doesn't refer to a specific number. Buuut, if you do the obvious thing and treat 0.(9) × 10 as "9. followed by an unspecified number of nines" the normal argument goes through.

Not sure how he'd reconcile that with the textbooks he likes to bring up though - this is literally examinable material (example attached)!

Enjoy.