Is $π/9$ an element of Cantor set on $[0,1]$?

Question

I know how to show that, for example $1/4$ is an element of the Cantor set using tertiary expansion but how do you use this method for an irrational number?

Answer 1

Exactly the same principle applies as in the rational case. You go down one digit in the ternary expansion at a time; if any digit is $1$ the number is not in the Cantor set. For $\frac\pi9$, the first digit is already $1$…

Almost all irrational numbers contain $1$ in their ternary expansions, and are thus not part of the set.

Answer 2

$\frac{\pi}{9}\in \left(\frac{1}{3},\frac{2}{3}\right)$, so it cannot be in the Cantor set.