In this paper, starting from recently known scaled hypercomplexes \({\mathbb {H}}_{t}\) , we define scaled hyperbolics \({\mathbb {D}}_{t}\) for scales \(t\in {\mathbb {R}}\) . In particular, the \(\left( -1\right) \) -scaled hyperbolics \({\mathbb {D}}_{-1}\) is isomorphic to the complex field \({\mathbb {C}}\) , the 0-scaled hyperbolics \({\mathbb {D}}_{0}\) is isomorphic to the dual numbers \({\textbf{D}}\) , and the 1-scaled hyperbolics \({\mathbb {D}}_{1}\) is isomorphic to the classical hyperbolic numbers \({\mathcal {D}}\) . For any fixed \(t\in {\mathbb {R}}\) , initiated from the t-scaled hyperbolics \({\mathbb {D}}_{t}\) , we construct the t-scaled-hyperbolic Clifford algebra \({\mathscr {C}}_{t}=\underrightarrow{\textrm{lim}}{\mathscr {C}}_{t,n}\) , where \({\mathscr {C}}_{t,n}\) are the n-th t-scaled-hyperbolic Clifford algebras for all \(n\in {\mathbb {N}}\cup \left\{ 0\right\} \) , with \({\mathscr {C}}_{t,0}={\mathbb {R}}\) and \({\mathscr {C}}_{t,1}={\mathbb {D}}_{t}\) , just like the classical Clifford algebra \({\mathscr {C}}={\mathscr {C}}_{-1}\) . To analyze this \({\mathbb {R}}\) -algebra \({\mathscr {C}}_{t}\) , we establish an operator algebra \({\mathscr {M}}_{t}\) (over \({\mathbb {C}}\) , as usual), containing \({\mathscr {C}}_{t}\) , and then construct a free-probabilistic structure \(\left( {\mathscr {M}}_{t},\tau _{t}\right) \) . From the operator theory, operator algebra and free probability on \({\mathscr {M}}_{t}\) , we apply these analysis for studying \({\mathscr {C}}_{t}.\)
