Let $x,y\in\ [0,1]$ a real numbers. Suppose that $$x=\sum_{n=1}^{+\infty}\frac{a_n}{3^n},\quad y=\sum_{n=1}^{+\infty} \frac{b_n}{3^n},\;a_n,b_n\in\{0,1,2\}$$ they are their ternary expansions.
Suppose we know only the ternary expressions and not actually the real number they represent.
Question Can I compare $x$ and $y$ working exclusively in base 3? Or rather, how can I determine if $x\le y$ or $y\le x$?
Thanks!
Yes you can, except for one small problem : some numbers have TWO different ternary developments.
Simple rule : if $a_n=b_n$ for all $n<n_0$, and $a_{n_0}<b_{n_0}$, then $x\le y$.
They can still be the same if $a_{n_0}=b_{n_0}-1$, and $a_n=2$ and $b_n=0$ for all $n>n_0$.