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the Theory of Triplets.
2
The extensive investigations show why Hamilton had to
consider a four dimensional Field extension by adding a new element
k
that is
linearly independent of the generators
1
,
i
, and
j
; and whose vector space forms the
basis
such that
i
2
j
2
k
2
1.
A quaternion is a hypercomplex number, which is an extension to the complex
numbers. The hypercomplex number domain possesses a “Numberlike” behavior in
R
n
and consists of symbolic expression of n terms with real coefficients, where n
may be 1 (real numbers), 2 (complex numbers), 4 (quaternion), 8 (cayley numbers
or octonion). These numbers can also be considered as Field extension of classical
algebra in hypercomplex number domain. They share many properties with com-
plex numbers with interesting exceptions. Quaternion algebra has all the required
properties except commutative multiplication, whereas the octonion algebra has all
the required properties except commutative and associative multiplication. They are
now used in computer graphics, computer vision, robotics, control theory, signal
processing, attitude control, physics, bioinformatics, molecular dynamics, computer
simulations, and orbital mechanics.
{
1
;
i
;
j
;
k
}
=
=
=−
2.2.2 Neurocomputing with Hypercomplex Numbers
The brief survey into the development of family of hypercomplex numbers point
out the fact that the idea of developing these numbers may generate normed division
algebras only in dimensions 1 (real numbers), 2 (complex numbers), 4 (quaternions),
and 8 (octonions). Hypercomplex numbers are direct extension of the complex num-
ber into the high dimension space. They can be seen as high dimensional vectors
comprising of components with one scalar and a vector in space. High-dimensional
neural networks developed using these numbers has natural ability of learning motion
in corresponding dimension because the unit of learning are these numbers (sig-
nals) flowing through respective neural network. The neural network in hypercom-
plex domain is an extension of the classical neural network in real domain, whose
weights, threshold values, input, and output signals are all hypercomplex numbers.
This chapter will clarify the fundamental properties of a neuron and neurocomputing
with hypercomplex number system.
2.2.3 Neurocomputing with Vectors
The word vector was originated from the Latin word vectus, which stands for “to
carry”. The modern vector theory was evolved from early nineteenth century when an
2
Interested readers may consult modern abstract algebra to understand difficulty of building a
three-dimensional field extension over
R
and Hamiltons breakthrough concerning the necessity of
three distinct imaginary parts along with one real.
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