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.

Sure, you can define anything you care to. In fact many commonly used systems, like decimal numbers, have elements with multiple representations like $0.999\ldots=1.$ Another example would be fractions, where $1/3=2/6.$
See A066352 in the OEIS for an example of an early use (by S. S. Pillai) of this particular system, and see also the related sequence A007924.
