Complex numbers in 6 dimensions

Abstract

Two distinct systems of commutative complex numbers in 6 dimensions of the polar and planar types of the form u=x_0+h_1x_1+h_2x_2+h_3x_3+h_4x_4+h_5x_5 are described in this work, where the variables x_0, x_1, x_2, x_3, x_4, x_5 are real numbers. The polar 6-complex numbers introduced in this paper can be specified by the modulus d, the amplitude ρ, and the polar angles θ_+, θ_-, the planar angle ψ_1, and the azimuthal angles ϕ_1, ϕ_2. The planar 6-complex numbers introduced in this paper can be specified by the modulus d, the amplitude ρ, the planar angles ψ_1, ψ_2, and the azimuthal angles ϕ_1, ϕ_2, ϕ_3. Exponential and trigonometric forms are given for the 6-complex numbers. The 6-complex functions defined by series of powers are analytic, and the partial derivatives of the components of the 6-complex functions are closely related. The integrals of polar 6-complex functions are independent of path in regions where the functions are regular. The fact that the exponential form of ther 6-complex numbers depends on cyclic variables leads to the concept of pole and residue for integrals on closed paths. The polynomials of polar 6-complex variables can be written as products of linear or quadratic factors, the polynomials of planar 6-complex variables can always be written as products of linear factors, although the factorization is not unique.