Biquaternion (Complexified Quaternion) Roots of −1

By Stephen J. Sangwine. 2005

61 citations recorded by the source corpus.

Abstract.The roots of −1 in the set of biquaternions (quaternions with complex components, or complex numbers with quaternion real and imaginary parts) are derived. There are trivial solutions (the complex operator, and any unit pure real quaternion), and non-trivial solutions consisting of complex numbers with perpendicular pure quaternion real and imaginary parts. The moduli of the two perpendicular pure quaternions are expressible by a single parameter by using a hyperbolic trigonometric identity.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b68809c8068212061b5462565dcacb7d9379ed27

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Square Roots of –1 in Real Clifford Algebras (2012, methodology): “Systematic research has been done [32] on the biquaternion roots of −1, abandoning the restriction to blades.”
Fundamental Representations and Algebraic Properties of Biquaternions or Complexified Quaternions (2010, methodology): “This section draws on and builds on results first presented by Sangwine in [33, 34].”
Convolution Theorems for Quaternion Fourier Transform: Properties and Applications (2013, methodology): “The QFT has been shown to relate to the other quaternion signal analysis tools such as quaternion wavelet transform, fractional quaternion Fourier transform, quaternionic windowed Fourier transform and quaternion Wigner transform [6, 16, 5, 3, 2, 27].”