Is it allowed to define a number system where a number has more than 1 representation? I was just curious; 
Is it allowed for a number system to allow more than one representation for a number?
For example, if I define a number system as follows:
The 1st digit (from right) is worth 1.
The 2nd digit is worth 2. 
The 3rd is worth 3. 
The 4th is worth 5.
The 5th is worth 7. 
The 6th is worth 11. 
And so on.... for all primes.
Now $9_{10}=10010_P$ for instance as $9=7+2$
But $8_{10}=1100_P=10001_P$.
Is that allowed?
P. S.
Is there any practical use of the number system mentioned above? Or not, since even operations like addition are almost impossible in it.

It is certainly allowed for a number to have multiple representations in a system of representations. In fact, the decimal system which we use all the time has this property. The well known fact that
$$1=0.999...=0.\overline{9},$$
is an example of this. Whether there is any practical use to the system you mention I don't know, but it seems unlikely. In the decimal number system every number has at most two representations, and it is easy to see whether two of them represent the same number. In your number system the number of ways to represent a number grows very quickly as the numbers grow, making it a nontrivial exercise to even figure out whether two representations represent the same number!
