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.

1. 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?
2. Similarly, is there a theoretical basis for saying that the floating point numbers are the best way to approximately represent real numbers?

Some thoughts:

• One could start with the following question: Given a finitely many letters, what is the most efficient way to encode integers?
• In case of real numbers, the question would be, given a finitely many letters (or just 0 and 1), how can we represent a dense subset of the interval (0,1) in the most efficient way? What about $\mathbb{R}$?