Quaternions Algebra by Using Brackets of Complex Numbers
Nasr Al Din Ide *
Department of Mathematics, Faculty of Science, Aleppo University, Syria.
Mehdi Jafari
Department of Mathematics, Faculty of Science, Aleppo University, Syria.
*Author to whom correspondence should be addressed.
Abstract
Quaternions and anti-quaternions, given by Hamilton in 1843, play an essential role in applied mathematics and in physics. In particular for calculating the rotation of a point in space Quaternions were studied by many researchers, which we will defined them by (2×2) matrices form of complex numbers, or as vector form of a real number and three complex numbers (as generalization of a complex numbers), or as (4×4) real matrix.
In this paper we give these quaternions in new form, which simplify the calculus, by using the brackets of complex numbers, then we study some property of these quaternions. And rotations in by using this method of brackets of complex numbers.
We study also various kinds of quaternions and investigate some of basic algebraic properties and geometric applications of them by using this method of brackets of complex numbers.
We see that, by writing a (4×4) real matrix in the form of brackets of complex numbers and by using the mathematical operations over these brackets, we found that the number of required operations in this case are less than the number of operations required in the case of classical matrices operations, in addition, this method simplifies and fast the calculations.
Keywords: Ecotypic variability, Generalized quaternion, Agronomic variation, rotation, Anthephora ecotype, split quaternion, quasi-quaternion, anti-quaternions