TITLE: Is there a mathematics field that studies the displaying of numbers?
QUESTION [5 upvotes]: I've read somewhere (not sure where) - that there is a dicotomy between the numbers and the symbols used for representing them, for example: We have the idea of twoness which can be represented in decimal base as $2$ and also in binary as $10$, in this case we're talking about the same number represented in two different bases. It could also be represented in roman symbols $II$ for example.
For the decimal base, there is a logic of succession, which is:
$$0,1,2,3,4,5,6,7,8,9$$
And then when we reach nine, we rotate the number in the left (which was $0$ previously):
$$\color{red}{\square} 8, \color{red}{\square} 9, \color{red}{1}0,...$$
The algorithm for roman symbols is fairly different:
$$I,II,III,IV,V,VI,VII,VIII,IX,X,...$$
It's possible to express $I$, $II$, $III$ and $V$, the four is formed with a rule of subtracting $1$ from the number in the right $\color{blue}{I}\color{red}{V}=\color{red}{5}-\color{blue}{1}=4$, It's the same for nine: $\color{blue}{I}\color{red}{X}=\color{red}{10}-\color{blue}{1}=9$.
Is there a field of study for these representations? Has anyone proposed axioms for their inner working or we've always used them intuitively? I remember of reading how these representations worked, but I never read about axiomatic basis for them nor sytems with completely different axioms.
Note: I have no idea on how to tag this question, if you know better tags, feel free to  edit.

REPLY [4 votes]: I think you are looking for numeration systems. In the book Automatic Sequences by Allouche and Shallit there is a chapter on numeration systems. From the introduction of Chapter 3:

In this chapter, we discuss how numbers can be represented by strings over a finite 
  alphabet. Our emphasis is on the representation of integers, although we briefly 
  discuss representations for real numbers in Section 3.4.
We start with the classical base-$k$ representation, and then discuss less familiar 
  representations such as representation in base $-k$ , Fibonacci representation, and 
  representation in complex bases.

I haven't had time to read this chapter yet, but it seems to contain a lot of information. 57 pages, 61 exercises and discussion of 8 open problems. This might just be the place to start if you want to know more.