On scaled hyperbolic numbers induced by scaled hyperbolic rings

In this paper, we generalize the well-known hyperbolic numbers to certain numeric structures scaled by the real numbers. Under our scaling of $\mathbb{R}$, the usual hyperbolic numbers are understood to be our 1-scaled hyperbolic numbers. If a scale $t$ is not positive in $\mathbb{R}$, then our $t$-scaled hyperbolic numbers have similar numerical structures with those of the complex numbers, however, if a scale is positive in $\mathbb{R}$, then their numerical properties are similar to those of the classical hyperbolic numbers. We here understand scaled-hyperbolic numbers as elements of the scaled-hypercomplex rings $\{\mathbb{H}_t\}_{t\in \mathbb{R}}$, introduced in [1]. This scaled-hyperbolic analysis is done by algebra, analysis, operator theory, operator-algebra theory and free probability on scaled-hypercomplex numbers