TL;DR: Meaning of these types of numbers: $1.2\overline{34}5\overline{67}$;
There exist (rational) numbers that are non-terminating, but have a repeating form of digits (e.g., $ 1.2\overline{34} $).
What I want to know is: does having 2 block of repeating digits make sense? My intuition says, "Why not?"; but my logic cannot comprehend, for example, when the 5 in the above example would occur and how you could, say, minus it from another number or w/e.
If so, can the resulting things be called numbers? What sort of numbers then? They wouldn't be rational, and probably not real; maybe we need a new (or an existing esoteric) system to be able to make them mean something and manipulate them.
Edit: Coming to think about it, another question that needs to be answered before considering this one is what is the meaning of numbers like $1.2\overline{34}5$. When is the $5$ reached? How would you go about trying to cancel that 5 or the repeating digits by minusing something?
Disclaimer: Excuse me if this question:
- has been already answered. I had not known the correct terminology and had not been able to to find anything on Google or MSE.
- does not make sense: As mentioned before, this may be because I am not able to express myself clearly.
- is not an answerable one or is irrelevant: If the answer is indeed "No," then I'd like a reason why so. After all, a lot of stuff was incomprehensible until systems to describe it were invented.
Update: Seems this sort of a question had been asked before right here on MSE, albeit by someone way more self-aggrandizing than me ;) causing it to get deleted. If you still want to see a copy of the question, check it out on the Wayback Machine.
Verdict: With no proper definition, rules for manipulation and useful model that they represent; these sort of numbers remain undefined and useless.
You edit, with $1.2\overline{34}5$ shows the problem well. The normal way to read this would be that the $34$ repeats forever and you never get to the $5$, in which case it is the nicely rational number $1\frac {116}{495}$. I wrote the below thinking you were looking at numbers like in the first line, where the repetition alternates between the $34$ and the $56$ with some changeable digit between them.
You can certainly conceive of numbers of the form $1.x34x56x34x56x34x56$, where the $x$'s anc represent any digit, not all the same. If they were all the same, we could just view $x34x56$ as one repeating set of digits and it is a rational like we know and love. If there is no pattern in the $x$'s, the number will be real irrational. Any given one of them can be operated with like any other real number.