For each integer $b\ge 2,$ we know that representations of real numbers are usually unique in the base-$b$ positional notation. The only time that uniqueness fails is if the form ends in a tail of $0$'s or a tail of $(b-1)$'s, in which case it is easy to convert between these dual representations. However, the fact that multiple representations are ever possible forces the mathematician to be additionally careful in writing some proofs. For example, in the standard application of Cantor's diagonal argument to show that the continuum is uncountable, one has to be careful to mention that we are constructing the rows using only terminating forms when there are dual representations and that the (anti-)diagonal element constructed is not somehow a dual form of one of the those terminating forms.
Question: Can a numeral system be constructed which represents all real numbers uniquely and only real numbers while still admitting some or all of the following convenient properties of the ordinary positional notation, and perhaps additional nice properties of its own:
- Being exponentially more efficient than unary, meaning the number of distinct integers represented by at most a certain number of digits is something like the number of distinct symbols in the system to the power of the number of digits.
- Admitting convenient pen-and-paper and computer algorithms for performing the arithmetic operations of addition, subtraction, multiplication, division and exponentiation, at least when integers or rationals are involved.
- Allowing for the existence of some convenient divisibility rules of integers, though not necessarily the same ones as those admitted by base-$b.$
- Having predictable (eg. periodic/cyclic) patterns in the representations of some large classes of real numbers, like the rationals.
If these properties are not possible to fulfill, I would still be interested in a system where there is uniqueness at the cost of losing these features. References to non-standard numeral systems that aim for such a goal (or perhaps other goals of convenience) would be appreciated.