Cantor's problems: the number $\xi =_{(3)} 0.222000222000...$ in $\mathbb{C}$ is a rational number $\frac{p}{q}$. Find $p$ and $q$.

Question

Let C be the Cantor's ternary set.

a) If $\xi =_{(3)} 0.02002000200002...$ is an element of C write which are the subintervals of $F_0, F_1, F_2, F_3, F_4$ and $F_5$ to which it belongs $\xi$.

b) The number $\xi =_{(3)} 0.222000222000...$ in C is a rational number $\frac{p}{q}$. Find $p$ and $q$.

I undertand the basics for the Cantor's set theory but I´m not really acquainted to this type of queations. Any hints would be great.

For a) I could do it by insepction but I belive there is better way to proceed.

Answer 1

$\dfrac{\xi}{27}=_{(3)}0.000222000222\ldots$

So $\frac{28}{27}\xi=_{(3)}0.222000222000\ldots+0.000222000222\ldots=_{(3)}0.222222222222\ldots=1$.

Therefore...

By the way, it is rather bold of you to use $\Bbb C$ to denote something other than the complex numbers!