Formulas for binary expansion of irrational number between $0$ and $1$

binaryclosed-formcomputabilityirrational-numbers

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?

Best Answer

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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