Modulo operation on fractions

Question

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 ideas are right, but the answers must be $(1\cdot 3)\bmod{13}=\color{red}{3}$, $(2\cdot 12)\bmod{19}=\color{red}{5}$ and $(4\cdot 2)\bmod{13}=\color{red}{8}$ respectively. — metamorphy, Oct 23 '18 at 09:06
@BillDubuque I see. Wasn't aware that it's a common practice. Thanks. — Siong Thye Goh, Oct 23 '18 at 13:59

score 1 · Accepted Answer · answered Oct 23 '18 at 09:13

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.

Modulo operation on fractions

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