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Similarity of Hybrid Numbers

- İskender Öztürk,
- MUSTAFA ÖZDEMİR

## Abstract

The set of hybrid numbers is a noncommutative number system unified and
generalized the complex, dual and double(hyperbolic) numbers with the
relation \(\mathbf{ih}\) \(=-\mathbf{hi}=\varepsilon+\mathbf{i.}\)
Two hybrid numbers \(\mathbf{p}\) and \(\mathbf{q}\) are said to be similar if
there exist a hybrid number \(\mathbf{x}\) satisfying the equality \(\mathbf{x}^{-1}\mathbf{qx}=\mathbf{p}\). And it is denoted by \(p\sim q\). In this paper,
we study the concept of similarity for hybrid numbers by solving the linear
equations \(\mathbf{px}=\mathbf{xq}\) and \(\mathbf{qx}-\mathbf{xp=c}\) for \(\mathbf{p},\mathbf{q},\mathbf{c}\in\mathbb{K}\mathbf{.}\)

Keywords : Hybrid numbers, quaternions, coquaternions, dual and
double numbers.

MSC Classification : 15A66, 11R52, 15A18, 15A06, 16S50.

29 Jan 2020Submitted to *Mathematical Methods in the Applied Sciences* 03 Feb 2020Submission Checks Completed

03 Feb 2020Assigned to Editor

15 Feb 2020Reviewer(s) Assigned

09 May 2020Review(s) Completed, Editorial Evaluation Pending

12 May 2020Editorial Decision: Revise Minor

19 May 20201st Revision Received

19 May 2020Submission Checks Completed

19 May 2020Assigned to Editor

19 May 2020Editorial Decision: Accept