I am not very good in math so I apologize if my question is too simple and does not belong here... Why an integer in numeral system X can be converted to decimal by multiplying it's digits to X (meaning base of the numeral system) which is raised to corresponding power (0,1,2,3 etc)? I mean that it seems like it is a universal way to convert any numeral system to decimal. But why there is no similar formula (involving multiplication by the base + exponents) to convert an integer of any numeral system to an integer in any numeral system? Thanks!
There is such a formula. There is nothing peculiar about base $10$. Suppose that you this number: $1\,332$, which is supposed to be a number written in base $4$. And suppose that you want to exprees it in base $6$. So, you compute $2$, $3\times4$, $3\times4^2$ and $4^3$, but you express them in base $6$. You'll get $2$, $20$, $120$, and $144$ respectively. Then you sum them (doing all computations in base $6$, and you'll get the answer: $330$.