I am not sure if I understood the modulo operation on fractions properly.
Therefore I want to ask if someone can provide me an intuitive visualization with an real-world analogy, in which that operation is can be applied (like the clock for the modulo operation on natural numbers)?
Further can someone please confirm, if the following equations are solved correctly?
\begin{align} \frac{1}{9}\mod 13 = 1\cdot9^{-1}\cdot (3\cdot9) \mod 13 = 1\\ \frac{2}{8}\mod 19 = 2\cdot8^{-1}\cdot (12\cdot8) \mod 19 = 2\\ \frac{4}{7}\mod 13 = 4\cdot7^{-1}\cdot (2\cdot7) \mod 13 = 4\\ \end{align}
The meaning for this fractions (and in fact for any fraction which is well defined) is: what is the inverse of...?
For example,$\;\cfrac19\mod{13}\;$ must be a number modulo $13$ that when multiplied modulop $13$ by $\;9\;$ you'll get $\;1\pmod{13}\;$ ...So $\;3\;$ works, since $\;9\cdot3=27=1\pmod{13}\;$ , and thus $\;\frac19=3\pmod{13}\;$
Follow this idea with the other ones.