I am trying to understand what normal numbers are. Just for simplicity I want to talk about base 10. I understand that a number is normal in base 10 if there a probability of $\frac{1}{10 } $ such that the numbers 0-9 pop up and a probability of $\frac{1}{100}$ that the numbers $0-99$ pop up in the decimal expansion and so on...
However I am wondering, in this case when we talk about normal numbers we are only talking about irrational numbers since numbers like $2$ or $3$ don't have a decimal expansion so there is no sense in talking about them as normal numbers?
Or for example a number like $\frac{1}{3} = 0.3333..$ is not normal base 10 since it's decimal expansion only contains the number 3.
I am just wondering if a normal number has to do anything with the normal form of a number.
As you noted a normal number ( in some basis) is necessarily irrational. We know (it was proved by E. Borel) that ''almost all'' (in the sense of measure theory) real numbers are absolutely normal normal, that is normal in all basis, but the proof is not constructive, and it is not clear if there is some computable normal number. What we can say is that we dont know if nubers as $\pi, e, \sqrt{2}$ are normal also in base $10$ .
We know that the Champernowne number is normal in base $10$ but it is not normal in all basis.
You can also see here for the problem of construction of a normal number.