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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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    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,](http://en.wikipedia.org/wiki/Champernowne_constant) which was proved transcendental by Mahler. It may be that Mahler's technique can be adapted.2012-04-02

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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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    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.2012-04-02