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.

You can certainly do this, and I can imagine situations where it might be useful.  For instance, suppose you take the normal decimal valuation of each place (ones, tens, hundreds, etc.), but allow "digits" from $0$ through $A$.  Then operations like addition and multiplication can be carried out with the usual algorithm, but with some flexibility about when/if to carry.  If you give me $999999 + 1$, the result could be $1000000$ or $99999A$... the latter requires fewer operations, and if I'm just going to subtract something from it in the future, might save a few steps.  As others have pointed out, comparing two numbers is no longer trivial; but it can be done by first coercing both numbers into a comparable form, e.g., performing all optional carries.  The relative efficiency has to depend on how often you're going to be comparing things vs. performing arithmetic operations.  (One thing you can say for sure, though, is that a non-unique representation takes more space, in bits, on average than a unique representation.  From an information-theoretic perspective, this is because you're encoding some useless extra data -- which representation you chose -- along with the actual value.)
