Exponentials get represented as a third dimension; where basic arthimetic is 1 or 2d depending on the context, exponentials go into a third dimension if that makes sense.
Modulo is the leftovers / splash-out when I pour one number into several smaller containers.
Fractions are simply fractional amounts of a tank of liquid (i.e. 2/3 is simply a measuring cup filled to the 2/3 line type of thing), but I can't ever picture them very accurately for weird fractions. "Improper" fractions are basically the same as modulo.. almost as if they're unstable in my head and automatically "pour" themselves into more tanks that fill as needed until some remainder is left.
I don't have a visualization for roots, which is probably why I'm generally so bad at them.
The representations helped in engineering school for getting a "feeling" about a formula; it was often very easy to notice if an equation I was massaging had gone off the rails. For a pure proof however (not that I did much of that), it was useless.
Your modulo visualization helps me, I think.
[1] - https://www.amazon.in/Visual-Thinking-Mathematics-Marcus-Gia...