Journal Homepage: www.ijrpr.com ISSN: 3049-0103 (Online) International Journal of Advance Research Publication and Reviews Vol 02, Issue 08, PP 809-819, August 2025 * Corresponding author. Tel.: 9412315836 E-mail address: jogendra.ibs@gmail.com Idempotent Elements in Tricomplex Numbers Jogendra Kumar 1 *, Anand Kumar 2 . 1 Department of Mathematics, Govt. Degree College, Raza Nagar, Swar, Rampur(UP), India. 2 Department of Mathematics, Govt. Degree College, Kant, Shahjahanpur(UP), India. * jogendra.ibs@gmail.com ABSTRACT In 1892, in search for special algebras, Corrado Segre published a paper in which he introduced an infinite family of algebras whose elements are called bicomplex numbers, tricomplex numbers… , n – complex numbers. In that paper Segre introduced idempotent elements of bicomplex numbers. Idempotent elements play a central role in the theory of bicomplex numbers. This paper introduces the algebraic structure of Tricomplex Numbers exploring some of their fundamental properties. We have identified and characterized sixteen distinct idempotent elements of Tricomplex Numbers. We also discuss their properties and establish the relationships among them. Keywords: Bicomplex Numbers, Tricomplex Numbers, Idempotent elements. 2010 AMS Subject Classification: 30G35, 32A30, 32A10, 13A18. 1. Introduction The set of bicomplex numbers is defined as (For detail cf. [1], [2], [3],[4], [5].): ℂ( 1 , 2 )={ 1 + 1 2 + 2 3 + 1 2 4 : 1 , 2 , 3 , 4 ∈ℂ 0 } where 1 ≠ 2 , 1 2 = 2 2 = −1 and, 1 2 = 2 1 . We shall use the notation ℂ 0 for the set of real numbers and ℂ( 1 ), ℂ( 2 ) for the following sets: ℂ( 1 )={ + 1 : , ∈ℂ 0 } ℂ( 2 )={ + 2 : , ∈ℂ 0 } Analogously, we also define ℂ( 1 , 3 ) and ℂ( 2 , 3 ). 1.1 Idempotent elements in ℂ( 1 , 2 ) Besides 0 and 1, there are exactly two non-trivial idempotent elements exist in ℂ( 1 , 2 ), denoted as 1 and 1 † and defined as 1 = 1+ 1 2 2 and 1 † = 1− 1 2 2 . Note that 1 + 1 † =1 and 1 1 † = 1 † 1 =0.