# Why multiplication by numeral system base and raising to corresponding power converts any numeral system integer to decimal only

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!

• Not sure this is clear. Do you mean something like $137=1\times 10^2+3\times 10^1+7\times 10^0$? If so, an analogous formula does hold for other bases. – lulu Jul 4 '18 at 16:10
• Yes, that exactly what I meant. But could you give an example of conversion from binary to duodenary system using the same approach? Thanks – user574077 Jul 4 '18 at 16:14
• Conversion between bases isn't as simple as all that. Think about decimal to binary, say. you need to find the largest power of $2$ less than or equal to the given number, subtract that and repeat. Simple, but not a closed formula. – lulu Jul 4 '18 at 16:24

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: $330$.