This is a direct follow-up to Conjecture on irrational algebraic numbers.

Take the decimal expansion for $\sqrt{2},$ but now think of it as the base $11$ expansion of some number $\theta_{11}.$ Is there an easy (or, failing that, hard) proof that $\theta_{11}$ is transcendental? Of course, same question stands for $\theta_k,$ for your favorite $k>10.$

istranscendental. But your question (whether there is an easy proof of it) is not answered by this. $\endgroup$ – Gerald Edgar Jul 6 '14 at 22:21certainsense, $\theta_{11}$ has a quicker rational approximations than $\sqrt 2$. Can this fact affect the irrationality measure of $\theta_{11}$? $\endgroup$ – Pietro Majer Jul 7 '14 at 12:42