What does the modulo of a non-integer mean?

Question

For example, in the equation

$ x=\frac{3}{5} \bmod 11$

The value of $x$ is $5$ according to wolfram alpha. I know how to manipulate the equation to to get the value but I dont understand what the modulo of a non-integer means.

Answer 1

In other words, $x=3\cdot 5^{-1}\pmod{11}.$ Here, $5^{-1}\pmod{11}$ indicates an integer $n$ such that $5n=1\pmod{11}.$ Do you know how to find such an integer?

Answer 2

The notation is ambiguous and requires context to interpret.

Additively $x=11n+\frac 35$ - the language of modulus is used in this way often with trigonometric functions mod $\pi$ or mod $2\pi$ and in similar contexts.

Multiplicatively it can be equivalent to $5x\equiv 3 \mod 11$.

In the absence of other information, with $11$ being prime, the most obvious interpretation would be the second, assuming that the context would be integer arithmetic modulo $11$, and noting that division is the inverse of multiplication, so the multiplicative interpretation is likely.