Improper bıcomplex numbers

Bicomplex numbers theory has been studied for a long time. The first chapter in this thesis, basic concepts such as homothetic motion, Lie groups, Lie algebra, matrix Lie group, curves, hypersurfaces, Dual numbers, Hamilton operator which are required in theory of the numbers are given. In the second part of the thesis, the set of bicomplex numbers is defined and subtraction and conjugate concepts are given. In addition, real matrix representations, complex matrix representations, the matrix Lie group structure, Lie algebra on hypersurfaces of M Lie group is generated and manifold structure of M is considered. In the third part of the thesis, the generalized bicomplex numbers are defined and conjugate concepts are given. Then, real matrix representation of generalized bicomplex numbers is obtained. Morever, Lie algebra of the generalized bicomplex numbers on hypersurfaces of M Lie group is generated. For the special case equalities in the bicomplex number are confirmed by using the generalization. Finally, the generalized dual bicomplex numbers are defined and conjugate concepts are mentioned.