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The number we are considering is as follows:

$0.a_1 a_2 a_3 \cdots $, where $a_{2n-1}=(n)_{(2)}, a_{2n}=(n)_{(3)}.$

So, the number is $$0.(1)(1)(10)(2)(11)(10)(100)(11)(101)(12)\cdots.$$ Is the number irrational? Is the number normal? Is the number transcendental?

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There are arbitrarily long strings of $2$'s, so the number is not rational. It is certainly not normal to the base $10$. –  André Nicolas Apr 2 '12 at 21:43
    
As to transcendentality, it is overwhelmingly likely (old joke, in various senses "almost all" numbers are transcendental.) As you probably know, your number is related to the Champernowne constant, which was proved transcendental by Mahler. It may be that Mahler's technique can be adapted. –  André Nicolas Apr 2 '12 at 22:00
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It's not rational because the decimals don't repeat -- far enough out there are arbitrarily long runs of decimals without any 2's, yet there are still infinitely many 2's.

It cannot be normal in base 10 either, because the limiting frequency if 7's in the decimal expansion is 0 where it should be 1/10 for a normal number. It might be normal in other bases.

Transcendental? Most likely, though I can't construct an argument for it right off the cuff.

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The usual argument for transcendence of the Champernowne's number (based on the Thue–Siegel–Roth theorem) applies here. –  Andres Caicedo Apr 2 '12 at 21:56
    
@Andres: If you know of an accessible source for that "usual argument", it would be interesting to have it added here. Neither Wikipedia nor Mathworld seem to link to one (apart from Mahler's original article, in German and probably only available on paper). –  Henning Makholm Apr 2 '12 at 22:08
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The best reference I know at an introductory level is: Making Transcendence Transparent, by E. Burger and R. Tubbs, springer.com/mathematics/numbers/book/978-0-387-21444-3 –  Andres Caicedo Apr 2 '12 at 22:38
    
As for the Thue-Siegel-Roth result, I am not sure an "elementary" treatment exists. I think a very good presentation can be found in: Diophantine Geometry, by M. Hindry and J. Silverman, springer.com/mathematics/algebra/book/978-0-387-98975-4 –  Andres Caicedo Apr 2 '12 at 22:41
    
My original intention was the number obtained from alternative mixture of general two base $k$ and base $l$ with $k,l$ distinct.Of course, first two questions are silly. Thanks for all the comments. –  hkju Apr 2 '12 at 23:23
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