Determining rationality of real numbers represented by prime digit sequence

[Jiketz]

TL;DR

The problem describes a sequence of real numbers formed by arranging prime digits in a certain pattern. If the original real number is rational, then all subsequent numbers formed from its decimal expansion are also rational. However, there exists an irrational number with the same pattern whose subsequent numbers are still rational.

I would like to know if my answer is correct and if no ,could you correct.But it should be right I hope:

 

[Mark44]

Thread locked. The OP has been advised to repost the question in one of the homework forum sections.