# Is it transcendental? Also normal?

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

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