In everyday practice the most common ways to represent integers are the binary and decimal systems. We use floating point or fixed point systems to (approximately) represent the reals. There are some other ways, not so widely used in computational sciences, such as continued fractions, and the use of the square root sign.
- Is there a way to formally assess the efficiency of these representations? Can we say, in some sense, the usual place-valued digital representation is the best way to represent integers?
- Similarly, is there a theoretical basis for saying that the floating point numbers are the best way to approximately represent real numbers?
The question is a big vague because the meanings of "efficiency" and "best" are also part of the question. Without these words, perhaps a reformulation would be: Do we have any justification to use the standard methods, beyond the fact that they are traditional, reasonably convenient, and we (arguably) do not know anything else to replace them yet?