On efficiency of notations for natural numbers

Computing a function on natural numbers assumes a fixed notation for natural numbers. Usually, we assume the binary or decimal notation. However, different notations might have different sets of effectively (PTIME) computable functions. We compare notations with respect to the set of effectively computable functions under a given notation. This approach gives an inclusion based partial ordering on notations, say ≤PTIME. We address the question whether the binary notation is, in some way, distinguished. We say that a notation σ is dense if the length of a σ-numeral for n is a function bounded by poly(log⁡(n+1)). We show that the binary notation is a maximal element in ≤PTIME among dense notations. We also show that a dense notation σ allows to compute effectively a different set of functions than the binary notation only if σ has short numerals for large numbers.