This question stems from an article,
"Entropy estimation of symbol sequences" download link
where the abstract mentions the need for using symbols in information theory.
Random chains of symbols (zeros and ones or characters) $s_1, s_2,\ldots,$ drawn from some finite alphabet appear in practically all sciences. Examples include spins in one-dimensional mag- nets, written texts, DNA sequences (here symbols are alphabets), geological records of the orientation of the magnetic field of the earth, and bits in the storage and transmission of digital data.
The paper presents technique for finding the entropy of symbols. My Question is that if entropy of symbols is important then what about the entropy of real valued numbers? From the perspective of entropy, is there a proof or some information in text where it provides information that the entropy of random symbols is greater or lesser than the entropy of real numbers.
Q1: In other words, what would be the entropy of random variables having binary values and entropy of real valued random variables? A fair coin takes 2 values -0/1 with equal probability and its entropy is 1 bit. So, is 1 bit the maximum entropy? Would entropy of binary random variables be less or more than entropy of real valued random variables and how to show?
So, I would like to know if there is a proof or material from which it can be inferred if the entropy of characters is lesser/ greater or same with the entropy of real numbers.
Q2: I want to know why symbols are used in coding and other applications such as storage from the viewpoint of entropy and information theory.