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

Question

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?

Answer 1

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.