• DOI:10.1080/0025570X.2004.11953236
• Corpus ID: 7837108

Geometry of Generalized Complex Numbers

@article{Harkin2004GeometryOG,
  title={Geometry of Generalized Complex Numbers},
  author={Anthony Harkin and Joseph B. Harkin},
  journal={Mathematics Magazine},
  year={2004},
  volume={77},
  pages={118 - 129}
}

• A. Harkin, J. Harkin
• Published 1 April 2004
• Mathematics
• Mathematics Magazine

Alternative definitions of the imaginary unit i other than i2 = −1 can give rise to interesting and useful complex number systems. The 16th-century Italian mathematicians G. Cardan (1501–1576) and R. Bombelli (1526–1572) are thought to be among the first to utilize the complex numbers we know today by calculating with a quantity whose square is −1. Since then, various people have modified the original definition of the product of complex numbers. The English geometer W. Clifford (1845–1879) developed the “double” complex numbers by requiring that i2 = 1. Clifford’s application of double numbers to mechanics has been supplemented by applications to noneuclidean geometries. The German geometer E. Study (1862–1930) added still another variant to the collection of complex products. The “dual” numbers arose from the convention that i2 = 0 [11]. Well known in kinematics is the use of dual number methods for the analysis of spatial mechanisms, robotic control, and virtual reality [4, 5, 10]. The ordinary, dual, and double numbers are particular members of a two-parameter family of complex number systems often called binary numbers or generalized complex numbers, which are two-component numbers of the form