Digits of irrational numbers

Voice_Of_Gods

Are there irrational numbers (but non-transcendental) which don't have a certain digit in their decimal expansion (for example, sqrt(2) has all the digits, from 0 to 9, in its decimal expansion)?

**Posted:** Thu Jun 10, 2004 3:46 am

Michael Lipnowski · Poincare Conjecture Offline Joined: 04 Mar 2004 Posts: 108

What's more interesting to me is whether $\sqrt2$ has infinitely many of each digit in its base $b$ representation, $b \geq 3$ .

If anybody knows anything about this, could you please post or give me a reference?

**Posted:** Thu Jul 15, 2004 6:41 pm

Magnara

On a related note, I was wondering if it is possible to, given a sequence of digits of finite length, find that sequence somewhere in the decimal expansion of irrationals and transcendentals. If the answer to your question is yes, then the answer to mine is clearly not true for all irrationals and transcendentals, but I wonder if it is true for any and if so, which ones.

**Posted:** Fri Jul 16, 2004 4:27 pm

rcv

**Posted:** Sat Jul 17, 2004 7:01 am

belenos · Riemann Hypothesis Offline Joined: 12 Sep 2003 Posts: 463

**Posted:** Sat Jul 17, 2004 7:45 am

Voice_Of_Gods

In my opinion, the number proposed by rcv is too articficialy constructed to be algebrical.

**Posted:** Sun Jul 18, 2004 4:21 am

lolo · New Member Offline Joined: 25 Oct 2004 Posts: 12

all of these are related to the still open Borel's problem :
"any algebraic irrational number should be normal in base b for any integer b>1 "
that means that any sequence of digits of lengh s should appears with the frequency 1/s in the base b expansion of such numbers.
Near nothing is known about that !

**Posted:** Sun Oct 31, 2004 2:11 am