## Properties of Quadratic Anticommutative Hypercomplex Number Systems

**Authors:** Katheryn Menssen

Hypercomplex numbers are, roughly speaking, numbers of the form x_1 + i_1x_2 + … + i_nx_{n+1} such that x_1 + i_1x_2 + … + i_nx_{n+1} = y_1 + i_1y_2 + … + i_ny_{n+1} if and only if x_j = y_j for all j in {1,2,…,n}. I define a quadratic anticommutative hypercomplex numbers as hypercomplex numbers x_1 + i_1x^2 + … + i_nx_{n+1} such that i_j^2 = p_j for all j (where p_j is a real number) and i_ji_k = - i_ki_j for all k not equal to j. These numbers have some interesting properties. In particular, in this paper I prove a generalized form of the Demoivre’s formula for these numbers, and determine certain conditions required for a function on a Quadratic Anticommutative Hypercomplex plane to be analytic—including generalizations of the Cauchy-Riemann equations.

**Comments:** 30 Pages. Professor Karen Shuman was my faculty advisor for this research project, which I did as an undergraduate. Although she helped greatly with editing throughout the process, all the research is my own ideas.

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### Submission history

[v1] 2019-12-09 11:43:58

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