An Introduction to Newly Defined Quaternions: Split-like Quaternions

In this paper, we define split-like quaternions such that p = p 0 + p 1 i + p 2 j + p 3 k, where i 2 = j 2 = -1, k 2 = 1 and p 0 , p 1 , p 2 , p 3 R. It is the purpose of this paper to describe a new non-commutative and non-associative quaternion algebra which is isomorphic to the four-dimensional Lorentz space R 4 1 .In addition, we define the inner product, characteristic of split-like quaternions, cross product and we discuss their properties.Furthermore, we give the matrix representations of split-like quaternions.Also, Cauchy-Schwarz inequality and triangle inequality provided by the inner product are demonstrated.Finally, we present the polar forms of split-like quaternions by giving the angle between two split-like quaternions and give De Moivre's formulas for split-like quaternions.