# Swapping the digits of an algebraic number (e.g. $\sqrt 2$)

Let an algebraic number, say $a=\sqrt 2 = 1.41421356237309504880...$, and define

$$b=f(a)=1.14243165323790058408...$$

by swapping the digits $a_{2i+1}$ and $a_{2i+2}$ for $i≥0$, corresponding to the decimal part of $a$ (we consider the decimal expansion, i.e. in base $10$).

My question is:

Is it true (or at least suspected) that $b$ is also algebraic?

It is not difficult to show that $y=f(x)$ is rational iff $x$ is, because a rational number has a finite or a periodic decimal expansion, and $f(f(x))=x$. This is why I chose an irrational algebraic number $a = \sqrt 2$.

Here are related questions : (1), (2), (3).

For instance, the number $a'=0.1010010001000010000010000001...$ is transcendental, and I believe that $f(a')$ is also transcendental.

Maybe this could be related to the (suspected) normality of irrational algebraic numbers, like $\sqrt 2$... I looked at Liouville's theorem, but I wasn't able to conclude.

Any comment will be appreciated !

• I see no reason to think that swapping digit would conserve a number's algebraic-ness. Swapping digit is such a random thing to do, and I'd say that $f(\sqrt{2})$ is almost certainly transcendental. – vrugtehagel Feb 15 '16 at 14:27
• @vrugtehagel : this is also what I think… but I don't know how it could be proven. I believe that this can be a difficult problem… – Watson Feb 15 '16 at 14:28
• Yes, unless there is an algebraic definition for this operation, it seems highly unlikely to be the case. Hard to prove, though, because it is hard to prove things aren't algebraic, in general. – Thomas Andrews Feb 15 '16 at 14:29
• More generally, you might ask: If $10x+y$ is algebraic, for $x,y$ real, then is $x+10y$? This is equivalent to asking whether $x-y$ is algebraic, which is thus equivalent to asking whether $x$ is algebraic. So you can rephrase whether taking every other digit $1.01020\dots$ is algebraic. – Thomas Andrews Feb 15 '16 at 14:33
• Another question, similar to this one, would be to know whether exchanging the digits $0$ and the digits $1$ (e.g. $0.1002301…$ becomes $0.0112310…$) preserves transcendence. – Watson Nov 1 '16 at 17:09