Exactly. Every instance of the belief that elements of some particular number systems are "real" is a mapping of the human abstraction representing that number

Exactly. Every instance of the belief that elements of some particular number systems are "real" is a mapping of the human abstraction representing that number system to another human abstraction of some other system of objects that are considered to be "actually real." The base case of that mapping is natural numbers representing possession of quantities of fungible objects. That mapping is broken by the negative integers, literally the next step up from the natural numbers. If you're concerned about the reality of the reals you can ramble on about finitism or the holographic principle and how something can possibly be real if you can't name or explicitly define every version of it, or you can recognize that every number system from the natural numbers to the complex numbers shares the property that elements of this system of numbers map to elements of physical systems and thus they are all exactly as real as each other.