# More than one blocks of infinitely repeating digits in a number

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:

• 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.

• 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 Sep 12, 2013 at 22:52

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.
• So are you saying that it is not a number? Why can't you remove the 5 by minusing $0.\bar{0}5$? Sep 10, 2013 at 3:03
• 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. Sep 10, 2013 at 3:11
• 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. Sep 10, 2013 at 14:54