Let

  √2 = 1 . d1 d2 d3 …     (di ∈ {0 , 1 , … , 9})

be the ordinary decimal expansion of √2 and, for an integer b ≥ 11, put

  θb := Σn≥1 di b-n. (1)

In words: we keep the same infinite string of decimal digits, but now regard it as the base-b fractional expansion of a new real number θb.
The question is whether one can prove, for some (or any) b ≥ 11, that θb is transcendental.

1. What can be proved easily
   • The digits of a rational number are eventually periodic in every integral base.
   Since the decimal expansion of √2 is not eventually periodic, the sequence (di) is not eventually periodic, hence θb cannot be rational. So θb is certainly irrational.
   • Apart from this, nothing essentially stronger is known at present.
   In particular no proof is known that θb is transcendental, and no proof is known even in the first case b = 11.

2. Why the known methods do not apply
   All classical transcendence proofs that work with “numbers given by their digits’’ (Liouville–type constructions, Mahler’s method, the use of continued–fraction patterns, automata–theoretic methods of Adamczewski–Bugeaud, etc.) require that the digit sequence possess some very explicit structure (rapidly growing gaps, functional self-similarity, finite automaton generation, …). The sequence of decimal digits of √2 is believed to be “random-looking’’; in any case no exploitable structure is known. Consequently all of the available techniques break down for θb.

3. Relation with famous open problems
   • Normality. It is not even known whether √2 is normal to base 10 (Borel’s 1909 conjecture). Proving transcendence of θb would be at least as hard as gaining substantial control on the digit distribution of √2, and thus appears to be out of reach at present.
   • Cobham-type results. If the digit sequence (di) were known to be automatic in some base, then Adamczewski–Bugeaud would imply transcendence of θb, but the digits of √2 are almost certainly not automatic – indeed, if they were, √2 itself would be transcendental, contradicting the quadratic equation it satisfies.
   • Simultaneous Diophantine approximation. One might hope to combine the good 10-adic approximations to √2 with the good b-adic approximations to θb and appeal to the Subspace Theorem; unfortunately the quality of the simultaneous approximations obtainable in this way is far too weak to violate known bounds for algebraic numbers.

4. General form of the problem
   Replace √2 by any fixed irrational algebraic number α and 10 by any base k ≥ 2 in which you write α. For every b > k (or, more generally, for every b multiplicatively independent of k) define θα,b = Σn≥1 dn(α,k) b-n , where dn(α,k) are the base-k digits of α. For no individual choice of α, k, b with α algebraic and b ≠ k is the transcendence of θα,b presently known.

5. Consensus
   It is universally believed that θb is transcendental (indeed, that θα,b is transcendental in the general setting), but this belief rests only on heuristic “random-digit’’ expectations and not on any theorem. At the moment the statement is completely open.

In brief: beyond the easy observation that θb is irrational, nothing is known – proving its transcendence would require a genuinely new idea in Diophantine analysis.