The System of R-Exp: Expanding the algebraic number system

Suppose we expand the algebraic number system as it is composed now by making a slight change to the rules of exponentiation?

What would that create?

It would open a new way to perform mathematical operations and fundamentally change the structure of algebra in general. Below is the article in which I propose to introduce addition as an exponential operation. Something I don’t think has ever been explored.

Additive Exponentiation