TITLE: What is meant, exactly, by nonrepeating when talking about irrational numbers?
QUESTION [22 upvotes]: My question is referring to the exact definition mathematicians use when describing the decimal expansions of irrational numbers as "nonterminating and nonrepeating." Now, I understand, at least ostensibly, what is meant by "nonterminating" and the phrase "nonrepeating" seems simple enough to understand, but I've always wondered what is meant by the exact definition:
It was always my understanding that the term "nonrepeating" was referring to a specific sequence of numbers showing up no more than once in the decimal expansion. I'm confused as to the exact criteria for fulfilling this requirement.

Surely it can't be just a sequence of $1$ number. In the sense that $\pi$ starts with the number $3$ and then the number $3$ shows up again, and again and again an infinite number of times.

Is it a sequence of $2$ numbers repeating then? For instance in the golden ratio $\phi = 1.61803398874989$ we could take any $2$ number sequence, say $61$ or $98$ or $33$ and would it be sufficient to say that that particular sequence never shows up again? That seems highly unlikely given the "nonterminating" nature of the decimal expansions for irrational numbers.

If not, then what sequence of $n$ numbers is sufficient to declare a number "nonrepeating?"

Moreover, philosophically, how does it make sense that any sequence of numbers doesn't show up more than once? When, necessarily, an irrational number has an infinitely long decimal expansion and a sequence of numbers (at least for practical determination) would be finite up to some $n \in \mathbb{N}$

I mean, the idea that the sequence length be infinitely long just seems like a convenient workaround that dilutes the significance of the "nonrepeating" quality of irrational numbers in the first place. Since, if you ever came upon a sequence that repeated for whatever $n$-digit sequence you had you could always just say "oh actually I meant this $(n+1)$-digit sequence instead!" and keep adding digits to the sequence ad infinitum.



Perhaps the term isn't referring to repeating sequences of numbers but rather the same numbers repeating one after another.

But this cannot be the case as we saw above with the Golden Ratio, in the short approximation written out we have $2$ cases where the same number is repeated immediately (i.e., $33$ and $88$) we also see this in this approximation for $\pi = 3.1415926535897932384626433$

So, if the term "nonrepeating" doesn't refer to repetition of sequences of numbers $n$ digits long, nor does it the consecutive repetition of the same number, then what else would it refer to?

REPLY [19 votes]: Others have provided a solid, formal definition, which is great. I'll post an answer that focuses more on the conceptual understanding of the thing.

It was always my understanding that the term "nonrepeating" was
referring to a specific sequence of numbers showing up no more than
once in the decimal expansion.

No. Repeating means that the decimal expansion at some point has a specific block of digits that repeats, one after the other, and nothing else ever happens. Nonrepeating means anything other than that.
And perhaps it bears highlighting that, logically, "not always repeating" (nonrepeating) is not the same thing as "always not repeating". It is, rather, equivalent to "sometimes not repeating". (Related question.)
I think some would say that the quality is rather obvious if you understand the mechanism that generates repeating decimal sequences. Take any fraction a/b. Start doing the long division: at each step there are only b possible remainders (specifically, the integers 0 to b-1). Because of this finite set of possible remainders, if the division doesn't terminate, then at some point it will hit one of the previously-seen remainders. And once that happens then you're literally repeating a step of division seen earlier in the process, which will start reproducing the exact same sequence of quotients and divisors after that point, forever.
A great exercise would be to actually do several of these long-divisions by hand and witness the repetition of digits as it occurs. Hopefully that experience will make the meaning of "repeating" very concrete.