Suppose we have an irrational number with the following decimal expansion:
$$A = a_0 \ a_1 \ a_2 \ a_3 \ a_4 \ a_5 \ a_6 \dots $$
Now, construct a new real number through a permutation on the decimals of the original number in the following manner:
$$A' = (a_1 \ a_0) \ (a_3 \ a_2) \ (a_5 \ a_4) \dots $$
Question 1: is $A'$ irrational, too?
We can generalize this question by considering other permutations as well. Let's consider the irrational number $A$ as above again. Let $\sigma_{k}(\cdot) $ be some permutation on a tuple of decimals of length $k$. Define
$$f_{k} (A) = \sigma_{k} (a_{0} \ a_{1} \dots \ a_{k-1} ) \ \sigma_{k} (a_{k} \ a_{k+1} \ \dots \ a_{2k-1} ) \ \sigma_{k} (a_{2k} \ a_{2k+1} \ \dots a_{3k-1} ) \dots $$
Question 2: is $f_{k} (A)$ necessarily irrational for all $k \geq 2$ and every possible permutation?