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On the hypercomplex numbers of all finite dimensions: Beyond quaternions and octonians

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posted on 21.03.2022, 14:36 by Pushpendra SinghPushpendra Singh, Anubha Gupta, Shiv Dutt Joshi
In search of a real three-dimensional, normed, associative, division algebra, Hamilton discovered quaternions that form a non-commutative division algebra of quadruples. Later works showed that there are only four real division algebras with 1, 2, 4, or 8 dimensions. This work overcomes this limitation and introduces generalized hypercomplex numbers of all dimensions that are extensions of the traditional complex numbers. The space of these numbers forms non-distributive normed division algebra that is extendable to all finite dimensions. To obtain these extensions, we defined a unified multiplication, designated as scaling and rotative multiplication, fully compatible with the existing multiplication. Therefore, these numbers and the corresponding algebras reduce to distributive normed algebras for dimensions 1 and 2. Thus, this work presents a generalization of $\mathbb{C}$ in higher dimensions along with interesting insights into the geometry of the vectors in the corresponding spaces.


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