We have been thought to notate a number n in base b as such, : e.g. 23 in base 10 $23_{10} =23$ , $3_2=11$ , n in base b as $n_b$
My problem is that both the number (n) and base (b) are written implicitly in base 10. is it possible to write the representation without using implicit base 10 , without ambiguity?
e.g. instead of $3_2$ to write $11_{10}$ without ambiguity? As motivation to this question, how can we be sure that $23_{10}$ is not actually what we consider as $23_7$ and $10_7$ (what ever the number 23 in base 7 is , and 10 in base 7 (i.e. 7 in base 10).
We don't need an implicit basis $10$, but we must write the number using the correct digits and the correct indication of the basis and, usually, this is done writing the base in base ten.
Your $3_2=11$ is wrong (there is not the symbol $3$ in base $2$). The correct way is $3_{10}=11_2$, or $3_{ten}=11_{two}$.