I am searching for some clues or solutions of the question below: For $\sqrt{2}=1.41421356\cdots$,
Is $1.1236\cdots$ irrational?
To say more formally: Let $f$ a function from $[0,1)$ to $[0,1)^2$ defined that for any real number x represented as $0.x_1x_2x_3\cdots$ in decimal (it is assumed that 9 never appears indefinitely now), $f(x)=(0.x_1x_3x_5\cdots, 0.x_2x_4x_6\cdots)$.
It is famous fanction as what is surjective but not injective, I wonder if $f(\sqrt{2}-1)=(0.1236\cdots,0.4415\cdots)$ has rational component.
This ploblem is not easy. If you don't think so, I want you to think about $f(0.040104020\cdots)=(\sqrt2-1,0)$
I thought many conditions for irrational numbers can't be used for this.
I'm sorry for my poor English.