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.
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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.
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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.
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If not, then what sequence of $n$ numbers is sufficient to declare a number "nonrepeating?"
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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?
Best Answer
The phrase "non-repeating" can be a bit confusing when first introduced. A more precise, if less snappy, term is "not eventually periodic" (and this is what mathematicians mean when they say "non-repeating" in the context in question).
A sequence of numbers $(a_i)_{i\in\mathbb{N}}$ is eventually periodic iff there are $m,k$ such that for all $n>m$ we have $a_n=a_{n+k}$. The "eventually" here is connected to the "$m$" - the sequence $$0,1,2,3,4,5,6,4,5,6,4,5,6,...$$ is not periodic but it is eventually periodic (take $m=4$ and $k=3$). On the other hand, the sequence $$0,1,0,0,1,0,0,0,1,0,0,0,0,1,...$$ is not even eventually periodic (although of course it does have lots of repetition in it).
The connection with irrationality is this:
(The issue re: these last two bulletpoints is that a few numbers have multiple decimal expansions. But this isn't a big deal to focus on at first.) In particular, the number $$0.01001000100001000001...$$ is irrational.
And base $10$, unsurprisingly, plays no role here: the above characterization works with "decimal expansion" replaced with "base-$b$ expansion" for any $b$.