The set of hybrid numbers 𝕂 is a noncommutative number system that unified and generalized the complex, dual, and double (hyperbolic) numbers with the relation ih  =−hi=ε+i. Two hybrid numbers p and q are said to be similar if there exist a nonlightlike hybrid number x satisfying the equality x−1qx = p. And, it is denoted by p∼q. In this paper, we study the concept of similarity for hybrid numbers by solving the linear equations px = xq and qx−xp = c for boldp,boldq,boldc∈𝕂bold.

Quaternions are an important tool that provides a convenient and effective mathematical method for representing reflections and rotations in three-dimensional space. A unit timelike split quaternion represents a rotation in the Lorentzian space. In this paper, we give some geometric interpretations of split quaternions for lines and planes in the Minkowski 3-space with the help of mutual pseudo orthogonal planes. We classified mutual planes with respect to the casual character of the normals of the plane as follows; if the normal is timelike, then the mutual plane is isomorphic to the complex plane; if the normal is spacelike, then the plane is isomorphic to the hyperbolic number plane (Lorentzian plane); if the normal is lightlike, then the plane is isomorphic to the dual number plane (Galilean plane).

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The set of hybrid numbers is a noncommutative number system unified and generalized the complex, dual and double(hyperbolic) numbers with the relation ih = −hi = ε + i. Two hybrid numbers p and q are said to be similar if there exist a hybrid number x satisfying the equality x −1 qx = p. And it is denoted by p ∼ q. In this paper, we study the concept of similarity for hybrid numbers by solving the linear equations px = xq and qx − xp = c for p, q, c ∈ K.

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This article aims to define the rotational motion around a lightlike axis in pseudo‐null or null frames more easily and to put forward a suitable number system to express rotation transformation in these frames. Therefore, for null and pseudo‐null frames, we define a new set of numbers consisting of a linear combination of two lightlike units and a spacelike unit, which we call Cartan numbers. The first part gives the properties of the set of Cartan numbers, and in the second part, some geometric definitions, theorems, and applications are given. In addition, the definition of the center of a parabola and the concept of the central angle are defined. The connection is made between these concepts and the parabolic rotation transformation. Rotation matrices for parabolic rotations are obtained. The rotating points indicated that it follows in which cases a line path and in which cases a parabolic path. The rotation transformation along any straight line or parabola is studied with examples.