When a rational series converges to an irrational number, can I operate on the sum of the rational series as I would a rational number?

Question

I am working on a problem involving Diophantine approximation. For some irrational value $0<\alpha<1$, I construct an infinite series $S = x + \frac{x_1}{10} + \frac{x_2}{10^2} + \frac{x_2}{10^3} + ... $ that converges to $\alpha$ where $x$ is the integer part of $\alpha$, and $x_1, x_2, x_3, ...$ denote the decimal values of $\alpha$. By fraction addition, we can determine the partial sum of S as $S_p = \frac{10^{p-1}(x) + 10^{p-2}(x_1)+ ... + x_p}{10^{p-1}}$. Let us define a number $\psi$ that for finite value $k$, $\psi>k$. In a sense, $\psi$ is transfinite. Define m as $$lim_{p\to\psi} \space S_p=m$$ Can I treat m as a rational number and operate on it as I would a rational number? (e.g. reduce numerator & denominator, compare it to a unit fraction, etc.)