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One can write any irrational number between $0$ and $1$ composed as closed expression of popular known numbers, such as, for example, the expression $$\frac{1}{\sqrt{2}}$$ in binary by successively dividing interval $[0,1]$ (each time in $2$) and comparing left right for each next bit to be determined.

One gets binary $$0.10110101000001001111001100110011...$$ for above example expression. But, I see no pattern, or do not know a closed formula or closed function to generate these particular bits (other than keeping on dividing and comparing).

Conversely, given a pattern or closed formula or closed function one can generate, i.e. compute, the corresponding irrational number (I assume the pattern is not simply repetition (for rationals)).

For example: binary $$0.101001000100001000001000000100000001000000001...$$ between $0$ and $1$ is constructed by each time increasing amount of $0$s between successive $1$s and equals (in decimal) $$0.641632560655...$$ a value which does not look to be a known expression of known numbers to me.

I wonder if perhaps any example exists, say something fancy like $$\frac{e}{\pi}$$ giving 'best of both worlds', that is: there is a pattern or closed formula or closed function to generate the binary expansion and the corresponding irrational number between $0$ and $1$ can also be written as closed expression composed of popular known numbers.

I am looking for such a, probably exceptional, binary expansion (in the context of probability) example but could not find one. Perhaps no such example exists?

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    $\begingroup$ "But, I see no pattern, or do not know a closed formula or closed function to generate these particular bits" Don't worry, nobody knows. $\endgroup$
    – jjagmath
    Commented Apr 1, 2022 at 23:13
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    $\begingroup$ Binary is no different from base $10$ in this regard. We don't have any way to find the base $10$ digits except by computing. Why should binary be different? $\endgroup$ Commented Apr 1, 2022 at 23:44
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    $\begingroup$ The series converge without regard to base. They don't necessarily make it easy to find the digits. For example, $\frac \pi 4=1-\frac 13+\frac 15 -\ldots$ does not refer to the base. It is an easy formula, but doesn't help you find digits in any base. $\endgroup$ Commented Apr 1, 2022 at 23:55
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    $\begingroup$ @RossMillikan I think you missed the point of the question. The OP is asking for an irrational number with both an explicit formula for the number and an explicit formula for the binary digits. $\endgroup$
    – jjagmath
    Commented Apr 2, 2022 at 1:22
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    $\begingroup$ My guess is that there is no such example. $\endgroup$
    – Peter
    Commented Apr 5, 2022 at 7:40

1 Answer 1

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Besides the rational numbers with a finite expression with continued fraction and a repetitive binary expansion, there are other numbers that have properties in both worlds.

The Rabbit Constant for example, can be define as $$\sum_{k=0}^\infty 2^{-\lfloor k\varphi\rfloor}$$ where $\varphi$ is the golden ratio.

It means that the binary expansion can be defined as the limit of the sequence of strings

  • $s_0=0$
  • $s_1=1$
  • $s_{n+1}=s_n\cdot s_{n-1}$ (string concatenation)

Hence obtaining the sequence (starting with "$0.$") $$101101011011010110101101101011011010110101101101011010110110101101101\cdots$$

But its infinite continued fraction is also $$[0;2^{F_0}; 2^{F_1}; 2^{F_2}; 2^{F_3}; \dots]$$ where $F_i$ are the Fibonacci numbers.

No known closed formula for this constant, but it's still very special to have simple expressions for those two very different systems beside rationals.

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  • $\begingroup$ Nice example too! $\endgroup$ Commented Apr 9, 2022 at 14:47
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    $\begingroup$ By lack of any even more spectacular example with nice properties I may want to soon mark this as solution :-) $\endgroup$ Commented Apr 10, 2022 at 1:26

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