A quaternion q [member of] H is a four-dimensional (4D)
hypercomplex number [11] and has a Cartesian form given by
Olariu [14] introduced the concept of
hypercomplex number and studied some of its properties.
Till date no similarity measure using hypercomplex number system in neutrosophic environment is available in literature.
In section 4 three dimensional hypercomplex number system and its properties have been discussed.
The
Hypercomplex Number of Dimension n (or n-Complex Number) was defined by S.
They cover complex and
hypercomplex numbers; octonion numbers; quaternions and color images; color images as two-dimensional grayscale images; one-dimensional and two-dimensional quaternion and octonion discrete Fourier transforms; color image enhancement and quaternion discrete Fourier transforms; gradients, face recognition, visualization, and quaternions; and color image restoration and quaternion discrete Fourier transforms.
Besides, in literature [32-34], the reformulation of incompressible plasma fluids and MHD equations has been discussed in terms of
hypercomplex numbers. Keeping in view the importance of quaternionic algebras, we establish the MHD field equations for dyonic cold plasma.
Hypercomplex Numbers in Geometry and Physics, 2004, v.
For example, one may well argue that "five" (the abstract quantity, fiveness) exists apart from us--even if humans had never evolved, you could still have five rocks in a field--and perhaps this extends to fractions and even irrational numbers; but when you start to talk about negative numbers, complex numbers,
hypercomplex numbers, infinitesimals, transfinites, matrices, vectors, multi-variable functions, tensors, fields, Galois groups, and the Mandelbrot set the compulsion to regard these as purely mental constructs is overwhelming.
Hamilton tried to deal with
hypercomplex numbers that could be presented as points in three or more dimensions.
Olariu [15] introduced the concept of
hypercomplex numbers and studied some of its properties in 2002, then studied exponential and trigonometric form, the concept of analyticity, contour integration and residue.
We can note here, for instance, that Yang-Mills field somehow can appear more or less quite naturally if one uses quaternion or
hypercomplex numbers as basis.