Quantum Mathematics?

Maybe I'm missing something, but wouldn't that just be grade school arithmetic? 1 doesn't exist between 0 and 1, because values between 0 and 1 are less than 1, and therefore not equal to 1. But 1 is between 1/2 and 1 1/2 because 1/2 < 1 < 1 1/2.
That would be called discrete math
Want to give my own take on this. It turns out "quantum mechanics" is a bad description for qm, since not everything is quantized there (you have continuous spectra all the time). Instead, mathematical study that include the name "quantum" in them tend to focus on the non-commutativity of qm. For example, in quantum probability you have non-commutativity of events. In quantum algebra the operators you apply to vector states are in general non-commutative.

So if you're looking for where the mathematicians are, I think "quantum algebra" isn't a bad start. That being said, it can include some more advanced abstract algebra that you'll want to be prepared for.
Quantization is part of geometry, but that's way beyond your league right now.

The mathematics that is used in quantum mechanics is just spectral theory which is part of linear algebra. You might need some PDE too.

There is also quantum logic, which mimic the nature of quantum mechanics.

There are also current research in rigorous nonperturbative quantum field theory. Since part of the research is making definitions, we don't even have a mathematical theory for it yet. Maybe one day.

All of them could be thought as "quantum mathematics".

>When I was little I was taught that a quantity of one does not exist between zero and one, but between one half and one & a half. There were several reasons, but mainly, the values following zero did not constitute an accurate representation of the whole function(?). I was being taught how to analyze "energy."

It's really hard to tell what you're being taught here. It sounds to me like a very basic fact, that number 1 is literally the middle point between 1/2 and 3/2, and not the middle point between 0 and 1. This fact would be so basic that there is not even a name for it.