In this paper, we consider split quaternions and split quaternion matrices. Firstly, we give some properties of split quaternions. After that we investigate split quaternion matrices using properties of complex matrices. Then we define the complex adjoint matrix of split quaternion matrices and we describe some of their properties. Furthermore, we give the definition of q-determinant of split quaternion matrices.

In this study, we will introduce arithmetical operations on dual hyperbolic numbers w x y ju jv      and hyperbolic complex numbers w x iy ju ijv     which forms a commutative ring. Then, we will investigate dual hyperbolic number valued functions and hyperbolic complex number valued functions. One can see that these functions have similar properties.

Müller [3], in the Euclidean plane E 2 , introduced the one parameter planar motions and obtained the relation between absolute, relative, sliding velocities (and accelerations). Also, Müller [11] provided the relation between the velocities (in the sense of Complex) under the one parameter motions inErgin [7] considering the Lorentzian plane L 2 , instead of the Euclidean plane E 2 , and introduced the one-parameter planar motion in the Lorentzian plane and also gave the relations between both the velocities and accelerations.In analogy with the Complex numbers, a system of hyperbolic numbers can be introduced: H := {x + jy | x, y ∈ R, j 2 = 1}. Complex numbers are related to the Euclidean geometry, the hyperbolic system of numbers are related to the pseudo-Euclidean plane geometry (space-time geometry), [5,15]. In this paper, in analogy with Complex motions as given by Müller [11], one parameter motions in the hyperbolic plane are defined. Also the relations between absolute, relative, sliding velocities (and accelerations) and pole curves are discussed.

Highlights• This paper focuses on the theories of dual-generalized and hyperbolic-generalized complex numbers.• The algebraic structures of dual-generalized and hyperbolic-generalized complex numbers are given.• Dual-generalized complex and hyperbolic-generalized complex valued functions are defined.• The matrix representations of dual-generalized and hyperbolic-generalized complex numbers are stated.• An efficient classification includes complex-generalized complex numbers is examined.

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