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?