Alagoz, Y.Ozyurt, G.2025-03-092025-03-0920240965-54251555-6662https://doi.org/10.1134/S0965542524700337https://hdl.handle.net/20.500.12662/4666The main aim of this paper is to introduce generalized quaternions with hyper-number coefficients. For this, firstly, a new number system is defined, which is the generalization of bicomplex numbers, hyper-double numbers and hyper-dual numbers. And any element of this generalization is called a hyper-number. Then, real matrix representation and vector representation of a hyper-number are given. Secondly, hyper-number generalized quaternions and their algebraic properties are introduced. For a hyper-number generalized quaternion, 4 x 4 real generalized quaternion matrix representation is presented. Next, because of lack of commutativity, for a hyper-number generalized quaternion, two different hyper-number matrix representations are calculated. Moreover, real matrix representations of a hyper-number generalized quaternion is expressed by matrix representation of a hyper-number. Finally, vector representations of a hyper-number generalized quaternion are given and properties of this representations are investigated.eninfo:eu-repo/semantics/closedAccesshyper-numbergeneralized quaternionhyper-number generalized quaternionArticle10.1134/S09655425247003372-s2.0-851961689599175Q390864WOS:001249179300004Q3