We shall understand the equation above as saying that each process, when applied to the element on the appropriate side of the equation, yields the element on the other side of the equation. But here we have not just any processes. First off, each of these processes undoes the other. Thus we view the eventually repeating decimal as encoding the fraction and vice-versa. One must interpret these equations carefully.
Similar remarks apply to the remaining equations.
Repeating the exercise above directly for decimal arithmetic would make for an awkward task. While the status of long division as an isomorphism stands behind all the math we do from grade school forward, actually proving it requires some new ideas.