I know that the binary and hexadecimal are useful, but what are the point of other bases, for example base 12? I know about the uses of binary and hexadecimal, but what are the uses of other bases, for example base 12? (or duodecimal)
 A: We use base 60 for time and for degrees. 
I don't know any practical use of base 12, but it would certainly be nicer than base 10. For instance, in base 12 the number 1/3 is not periodic: you have $$1/3=4/12=4\times 12^{-1}=0.4_{12}.$$
A: Conversion between different bases turns out to be a useful way to talk about various pathological functions in analysis.
Some examples of this:


*

*The Cantor function $c$: the base-2 expansion of $c(x)$ is closely related to the base-3 expansion of $x$.

*The Conway base-13 function $f$: the base-$10$ expansion of $f(x)$ is closely related to the base-$13$ expansion of $x$ (or, more generally, you could have a version of it where the base-$k$ expansion of $f(x)$ was closely related to the base-$(k+3)$ expansion of $x$, for any $k \geq 2$).
A: The theory for developing and computing the binary or hexadecimal representation of a number applies equally well to other bases. They may or may not be useful. If some species of alients have 12 fingers they may find base 12 useful.
Having too large base has one practical difficulty: finding names for the numbers and remembering them. Too small a base makes even a moderately small number requiring large number of digits to write down.
Civil engineers may have the right math that tell us how deep a foundation has to be to build a 500-storey building. Will we build it? 
A: Their use is precisely the use you have for base $10$: they provide a means of representing numbers concretely. The only difference between  $17_{10}$ and $10_{17}$ is that you are accustomed to the former representation rather than the latter to represent the number of periods in the quoted symbol ".................".
