IJPAM: Volume 82, No. 1 (2013)


John Lanta$^1$, Kenneth K. Nwabueze$^2$
Department of Mathematics and Computer Science
Papua New Guinea University of Technology

Abstract. The utility of complex analysis in providing a rich family of holomorphic functions has motivated the extension of the planar theory based on complex numbers to a $4$-space study with functions of a quaternion variable. A function of a quaternion variable is a function with domain and range in the quaternion. Such functions naturally appear when one considers the projection of a quaternion onto its scalar part or onto its vector part, as well as the modulus and versor functions. A very useful function of quaternion variable is the function $f(v) = q v q^{-1},$ which rotates the vector part of a quaternion $v$ by twice the angle of a quaternion $q.$ In this paper, we present a brief survey of some interesting discrete properties of $f(v)$ as well as propose its utility in the description of pythagorean triples. Representing the quaternion in matrix form, along with a microscopic modification of our results in this paper, resolves an open problem by Mircea Crasmareanu [#!CM!#] concerning a connection between the pythagorean triple preserving matrix and the algebra of quaternions.

Received: August 13, 2012

AMS Subject Classification: 11D09, 20D15

Key Words and Phrases: complex number, quaternion, rotation, pythagorean angle, pythagorean triple

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2013
Volume: 82
Issue: 1