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=
idea works: n
1, 2, 4, 8. These number systems have many common algebraic and
geometric properties.
A literature survey into the vector algebra brings out the fact that many searches
had gone to enable the analysis of quantities, which involve magnitude and direc-
tion in three-dimensional space, in the same way as complex numbers had enabled
analysis of two-dimensional space, but no one could arrive at a complex numbers
like system. Therefore, another way to represent a vector in three dimensional space
(or
R
3
) are identified with triples of scalar components. These quantities are often
arranged into a real-valued vectors, particularly, when dealing with matrices. These
standard basis vectors can also be generalized into n-dimensional space (or
R
n
).
This chapter is devoted to present a theoretical foundation of hypercomplex num-
ber system and high dimensional real-valued vector. This chapter also establishes
their basic concepts for vector representation along with their algebraic and geomet-
ric properties in view of designing high dimensional neural networks. The successive
chapters of the topic lead to their vital applicability in various areas.
2.2.1 Hypercomplex Number System
The nineteenth century is observed as the very thrilling time for philosophy of
complex numbers. Though, complex numbers had been discussed in works pub-
lished in the sixteenth century, the study of complex numbers was totally dismissed
as worthless at that time. After three centuries the sensibleness of complex numbers
was truly understood when most of the fundamental results, which now form the core
of complex analysis, were exposed by Cauchy, Riemann along with many others.
Even Irish mathematician and physicist William Rowen Hamilton was fascinated by
the role of complex number system in two-dimensional geometry in the nineteenth
century before discovering quaternions. Indeed, set of real numbers '
R
' is a sub-
set of set of complex numbers '
C
';
C
is a Field extension over
R
. It is a number
system where we can add, subtract, multiply, and divide. The algebraic structure
“doublets” or “couplets” (
a
R
2
) was regarded as a algebraic representation
of the Complex Numbers, which can easily use complex arithmetic to do various
geometric operations. The Field of complex numbers is defined by
+
ib
∈
i
2
C
={
a
+
ib
|
a
,
b
∈
R
;
=−
1
}
The field of complex numbers is a degree two field extension over the field of real
numbers; the vector space of C forms the basis
{
1
,
i
}
over
R
. This means that every
complex number can be written in the form
a
+
i
b
, where a and b are real numbers
and i is an imaginary unit (
i
2
1). Mathematicians want to construct a new fields
for hypercomplex numbers such that
C
becomes a subset of hypercomplex number
system, and the new operations in them are compatible with the old operations in
C
. Therefore, mathematicians were looking for a field extension of
C
to higher
dimensions, hence hypercomplex numbers.
=−
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