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I recently made an edit to this post concerning $\pi$ and it containing all possible combinations of numerical values; and this answer to it brought forward an interesting number:

0.011000111100000111111…

This got me thinking; what is it called when a number has a pattern that can be replicated infinitely, though the same number never repeats. The best example is the above referenced nuber; when this is broken down:

0, 11, 000, 1111, 00000, 111111

Granted even this is not a perfect example as zero and one are repeated which breaks the same number never repeats rule if you take it that far; this would mean that further definition is required.

I suppose a thorough definition would be more of:

A number whose digits represent a pattern that can be scaled infinitely, without repeating grouped digits such as:

10110111 - zero repeats, not a true resemblance.

011000111100000111111 - zeros are grouped, true resemblance.

The Question at Hand: what is it called when a number has a pattern that can be replicated infinitely, though the same number never repeats.

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Hazel へいぜる
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    There is no standard term for this - it would require defining what a "pattern" is and isn't, which can be tricky... – Jair Taylor Aug 13 '18 at 20:00
  • Is it safe to assume that the trickiness you speak of is related to the pattern would have to be defined either generically or defined for each pattern? – Hazel へいぜる Aug 13 '18 at 20:01
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    One might argue that every number's decimal expansion follows a certain "pattern." After all, every time you try to calculate $\pi$ to a certain degree of accuracy, the decimal digits are always going to wind up the same. The first digits of $\pi$ always begin $3.14159265\dots$. How does the "pattern" that you find in the digits of $\pi$ fundamentally differ from the "pattern" you see in the number you describe? Why is one "pattern" more "pattern-esque" than the other? – JMoravitz Aug 13 '18 at 20:19
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    I can think of two formal, objective, common definitions which may capture the idea of "having a pattern". One is that we eventually come across an infinitely repeating pattern, which gives a rational number. The other is that the pattern can be exactly described using finitely many words (with a formal requirement on what words and expressions are allowed) and any given digit can, using that description, be calculated in a finite amount of time, which gives you the computable numbers. Anything in-between would probably be pretty arbitrary. – Arthur Aug 13 '18 at 20:25
  • I've updated my question with a more descriptive definition in the hopes that maybe this would be a more realistic definition. I may need help with the wording here. – Hazel へいぜる Aug 13 '18 at 20:34
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    There's no name for it. Usually, we name things when we have cause to refer to them frequently; what's so special about such decimal expansions that warrants their having a name? Note, for example, that rationals are not named for their decimal expansions (which is too dependent on the number $10$) but for the fact that exactly two integers can be used to define them. So, why would you want to give such series as $0.1234567891011121314151617181920...$ a name? – Allawonder Aug 13 '18 at 20:41
  • "...which breaks the 'same number never repeats' rule if you take it that far..." I do not understand what you are getting at here. Surely, you are interested specifically in irrational numbers with a specific property and so are avoiding some number (or specific sequence of numbers) consecutively repeating without any other number inbetween (thereby ensuring the number is indeed irrational and not rational). Your number you describe is indeed irrational and does not have any point where you have "infinitely many ones" back to back, for example. – JMoravitz Aug 13 '18 at 20:41
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    Regardless, I think the clear answer to the original question "what are these called?" will eventually be either that there is no name for them, or they are the computable numbers. I think the better question to be asking is "how might we formalize the concept of 'having a pattern?'" – JMoravitz Aug 13 '18 at 20:43
  • @Allawonder I wouldn't specifically give just any irrational number a name; I'm thinking more along the lines of a category of numbers that can all be expressed the same such as irrational numbers not being able to be expressed as fractions. – Hazel へいぜる Aug 13 '18 at 20:45
  • @ JMoravitz ,Here are some possible notions of "pattern" in this sense

    We say that the digits of x have a weak pattern in base b if there's a polynomial time algorithm in n to compute the nth digit in base b.

    Note that a weak pattern is very weak and may not even be obvious. It may make sense to use a more narrow computational complexity class, like say insist that the calculation time be O(n). Another option would be to declare that the digits should correspond to an automatic sequence https://en.wikipedia.org/wiki/Automatic_sequence .

     – JoshuaZ Aug 13 '18 at 21:05
  • "infinitely repeating pattern, which gives a rational number" I .. disagree. An infinite periodically repeating pattern gives a rational number. But I'd say a number like $101001000100001....$ = "add a zero each time" can be said to be "infinitely repeating"-- it's a simple rule that ... repeats. The question is how to formalize and say "that is a pattern" whereas "calculate pi" is not. – fleablood Aug 13 '18 at 21:18
  • "So, why would you want to give such series as 0.1234567891011121314151617181920... a name?" Well, "because they are neat looking" is as good a reason as any. I think it's important to realize we did not give period repeating numbers the name rationals. We name ratios of integers rationals (an more importantly those that are not irrationals). The observation that they are repeating patterns is a consequence and not a definition. – fleablood Aug 13 '18 at 21:26
  • @fleablood How does their being "neat-looking" (whatever that means; is the decimal expansion of $π$ neat-looking? If so, how is this class of numbers different from the irrationals, if not why not?) require their being named? In particular, do neat-looking irrationals have an interesting property behind them? Do they reveal a deeper structure? Do they form a "nice" subset of $\mathbf R$? These are the sorts of questions that make something worth a name, not just because it happens to satisfy our whim; otherwise one could name every single mathematical oddity, which is a boring business indeed – Allawonder Aug 13 '18 at 22:29
  • I never said being "neat looking" required a name. I said that being "neat-looking" was as good a reason as any for giving them a name. " In particular, do neat-looking irrationals have an interesting property behind them?" Yeah... they are neat looking. – fleablood Aug 14 '18 at 00:22
  • "otherwise one could name every single mathematical oddity" Um... yeah?? "which is a boring business indeed" Um,.... no. – fleablood Aug 14 '18 at 00:23

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There's no real term for it.

It's an irrational number though.

We refer to "patterns" but what we really mean and are interested in is "periodic". For a periodic decimal, there is a point where every $k$th decimal term repeats; that is to say, for large enough $j$ then the $j$th decimal, $a_j$, will be equal to the $j+k$the decimal, $a_{j+k} = a_j$.

The only reason we are interest in that type of pattern is because that means the value itself is rational.

All numbers are either rational, can be written as $\frac mn$ where $m$ and $n$ are integers. Tho write the decimal of $\frac mn$ there are only so many possible remainders so we must repeat remainders eventually. That leads to an infinite loop with a periodic repeating. Likewise if we have a periodic repeat of period $k$ and we multiply by $10^k - 1$ we get something that terminates so it must be rational.

So we have the very useful result: An number is rational if and only if it's decimal expansion is periodic.

Or to make the language to high school students simpler and not intimidating: "if the decimal has a pattern".

So the pattern you describe is ... interesting and probably be worth studying. But algebraically it doesn't have any significance, in and of itself.

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Post-script: It's important to realize "decimal numbers that repeat periodically are rational" is a consequence; not a definition. (They are ratios of integers and the periodic repeating is just a consequence.) Here "incremental patterns" are number with predictable patterns such as $.101001000100001000001.....$ are a definition itself.

fleablood
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  • Thank you highly for your answer! I wouldn't say numbers such as the one I quoted or any other similar numbers are insignificant, we just don't currently have any reliable uses for them yet. For example, $\pi$ always existed and only became significant once it was discovered that $\pi$ * d was the circumference of a circle. Just look at some of the calculation methods for $\pi$ or *e*. – Hazel へいぜる Aug 13 '18 at 21:26
  • I did not say they were insignificant. I said that in terms of their algebraic structure the fact they have nonrepeating describable patterns doesn't have any inherent significance other than the pattern. We can ask "what are the numbers whose decimals contain only odd digits?". We don't call them anything. Their significance is in an d of itself simply that the contain only odd digits.I wouldn't say that $\pi$ "always existed" or "became significant". No-one said "hey you know that number $\pi$ that we keep tripping on? I just found out it's the ratio the circumference to diameter ratio!" – fleablood Aug 13 '18 at 21:36
  • I would say that discovery was probably more along the lines of: "I have proven that the ratio of any circumference to its diameter is $\pi$.". Of course supplying the actual number (to its accuracy at the time) and eventually people grew tired of writing out and memorizing the decimal places, and began utilizing the $\pi$ representation at some point. – Hazel へいぜる Nov 12 '18 at 17:30
  • Oh, absolutely not! The concept that the ratio and concept had to exist first. And the symbol existed before the concept the decimals were known. No-one ever refered to it by decimals. In fact decimals themselves are a very new concept. Confusing knowing what a number is with knowing it's decimal is a naive and common mistake but it is a mistake. – fleablood Nov 13 '18 at 05:19