6

Are all normal numbers transcendental?

Just a question I've come up with.

user89167
  • 785
  • 1
    "Simply normal in base $10$" is even compatible with rational: $0.\overline{1234567890}$, but that is of course a lot less than "normal" (to any base). - I don't see however, which pattern should occur unusually often in $\sqrt 2$ – Hagen von Eitzen Apr 13 '15 at 17:25
  • @HagenvonEitzen "Simply normal" is not very interesting -- it only says every digit appears with probability $\frac{1}{10}$. What is interesting is how a normal number contains every single digit combination. – user89167 Apr 13 '15 at 17:31

1 Answers1

4

I found this on Wikipedia:

It has been conjectured that every irrational algebraic number is normal; while no counterexamples are known, there also exists no algebraic number that has been proven to be normal in any base.

So while this does not answer your queston, it shows that the answer is not that simple ...

Hagen von Eitzen
  • 374,180