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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} $).

  1. 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.

  2. 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.

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  • $\begingroup$ While the numbers you're looking for don't really make sense, the idea of a repeating block, and then some things, and then another repeating block isn't nonsense - it just doesn't match up with standard decimal notation. See my answer to a different question for an application of that idea to a different sort of number system math.stackexchange.com/a/483582/26369 $\endgroup$
    – Mark S.
    Sep 12, 2013 at 22:52

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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.

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  • $\begingroup$ So are you saying that it is not a number? Why can't you remove the 5 by minusing $0.\bar{0}5$? $\endgroup$ Sep 10, 2013 at 3:03
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    $\begingroup$ You need to define what you mean by $1.2\overline{34}5$. We have a definition of $1.2\overline{34}$ as a limit: it is $1.2+\frac {34}{10^3}+\frac{34}{10^5}+\dots$ The question is how does the $5$ change that? Since the $34$'s go on forever, you don't get to the $5$, so it doesn't matter. In that spirit, you would never get to the $67$ in the original problem, so it is the same number. I read it differently, finding a way to pay attention to the $67$, but it is on you to say what you really mean. In a sense, I responded to $1.2\overline{34x56x}$. If that is not what you meant, say so. $\endgroup$ Sep 10, 2013 at 3:11
  • $\begingroup$ That is not what I meant; I had meant an infinitely repeating block of digits, followed by another infinitely repeating set of digits. But what I think you're trying to say is that since anything after the first set of infinitely repeating digits is never really reached, they are negligible and useless to consider; am I ryt? $\endgroup$ Sep 10, 2013 at 14:18
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    $\begingroup$ That is correct. A good way to think of an infinite decimal is as a series. If the digits are $d_i, i\in \Bbb N$, the number is $\sum_{i=1}^{\infty} \frac {d_i}{10^i}$. This is a convergent series and converges to the value. Since the first repeat provides all the digits you need, you never get past it, so nothing else matters. $\endgroup$ Sep 10, 2013 at 14:54
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    $\begingroup$ Well, I guess that sums it all up then. With no proper definition, rules for manipulation and useful model that they represent; these sort of numbers remain undefined and useless. $\endgroup$ Sep 11, 2013 at 12:23

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