Is there a math field that studies something like this?

I was having a blurry thinking of differences about rational and irrational numbers, then I had the idea of ploting them in a specific way:

$$\frac{1}{2}=0.5$$

Getting that value, I've thought about a function for the digit positions, which in the given case would give me

$$\begin{matrix} {f(1)}&=&{0}\\ {f(2)}&=&{5} \end{matrix} $$

And in the case of

$$\frac{1}{3}=0.33333...$$

Would give me

$$\begin{matrix} {f(1)}&=&{0}\\ {f(2)}&=&{3}\\ {f(3)}&=&{3}\\ {f(4)}&=&{3}\\ {f(5)}&=&{3}\\ {f(n>5)}&=&{3}\\ \end{matrix} $$

I've even plotted some examples:

Is there a math field that studies something like that? The nearest thing that comes to my mind are the Ford Circles (although I know they're very different). In the Ford circles, there is a relation between the radius of the circle and the number it represents. I was thinking also if such a relation could be found in this?

I'm not thinking about a relation exactly like Ford circles, I'm thinking about a deeper relation between the line and the given number.

Some fields that occur to me are:

1. Number theory
2. Normal numbers
3. Summations for $\pi$ and other constants
4. Digit-extraction algorithms (for our good friend $\pi$ and other constants
5. Periods of the reciprocals of primes (and other integers) in different bases

Linear algebra: Represent the numbers as polynomials (assuming x is 0.1, or some positive integer exponent of I guess) in a polynomial vector space. Let its domain be the natural numbers. Let f(n) be the inner product of said polynomial with the $n^{th}$ monomial basis vector.

You can probably do some funky stuff with this through the lens some part of more general abstract algebra or algebraic number theory.