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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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    $\begingroup$ There is no standard term for this - it would require defining what a "pattern" is and isn't, which can be tricky... $\endgroup$
    – Jair Taylor
    Aug 13, 2018 at 20:00
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    $\begingroup$ 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? $\endgroup$
    – JMoravitz
    Aug 13, 2018 at 20:19
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    $\begingroup$ 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. $\endgroup$
    – Arthur
    Aug 13, 2018 at 20:25
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    $\begingroup$ 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? $\endgroup$
    – Allawonder
    Aug 13, 2018 at 20:41
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    $\begingroup$ 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?'" $\endgroup$
    – JMoravitz
    Aug 13, 2018 at 20:43

1 Answer 1

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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.

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  • $\begingroup$ 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. $\endgroup$
    – Hazel へいぜる
    Aug 13, 2018 at 21:26
  • $\begingroup$ 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!" $\endgroup$
    – fleablood
    Aug 13, 2018 at 21:36
  • $\begingroup$ 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. $\endgroup$
    – Hazel へいぜる
    Nov 12, 2018 at 17:30
  • $\begingroup$ 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. $\endgroup$
    – fleablood
    Nov 13, 2018 at 5:19

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