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As next natural step, Hamilton was eager to extend the complex numbers to a new
algebraic structure with each element consisting of one real part and two distinct
imaginary parts, which would be known as “Triplets” and forms the basis
.
There are necessary and sufficient mathematical reasons, why one should attempt for
such a construction. He desired to use these triplets to operate in three-dimensional
space, as complex numbers were used to define operations in the two-dimensional
plane. He attempted for years to invent an algebra of triplets
{
1
,
i
,
j
}
R
3
(
a
+
i
b
+
jc
∈
)
to
play same role in three dimensions.
Can triplets be multiplied? Hamilton worked unsuccessfully in creating this struc-
ture for over 10 years. After a long work he observed that they can only be added and
subtracted; Hamilton could not solve the problem of multiplication and division of
triplet. We now know that this pursuit was in vain. Historically, it was noted that on
October 16th, 1843, while walking with his wife along the Royal Canal in Dublin,
the concepts of using quadruple with the rules of multiplications dawned on him.
Hamilton discovered a four dimensional algebraic structure called the quaternions.
The discovery of the quaternions is one of the most well documented discoveries in
mathematics. In general, it is very rare that the date and location of a major math-
ematical discovery are known. Hamilton explicitly stated, “I then and there felt the
galvanic circuit of thought and the sparks which fell from it were the fundamental
equations between
i
,
j
,
k
; exactly such as I have used them ever since in complex
number system”.
The set of quaternions, often denoted by
H
in honor of its discoverer, constitute a
noncommutative field (a skew field) that extends the field
C
of complex numbers. The
quaternions is constructed by adding two new elements
j
and
k
in complex number,
thus new algebraic structure would require three imaginary parts along with one real
part. For this new structure to work, Hamilton realized that these new imaginary
elements would have to satisfy the following conditions:
i
2
j
2
k
2
=
=
=
ijk
=−
1
One could now talk about the additive and multiplicative operations that can be
defined on elements of H and turn it into a field. The field of quaternions can then be
written as
and
i
2
j
2
k
2
H
={
q
=
q
0
+
q
1
i
+
q
2
j
+
q
3
k
|
q
n
∈
R
;
=
=
=
ijk
=−
1
}
If Hamilton had been able to develop his Theory of Triplets, he would have
effectively built a degree three field extension of
R
whose vector space forms the
basis
over
R
such that
i
2
j
2
1. This field must be closed under
multiplication. After struggling with all possibilities, researchers including himself
came in conclusion that there is no third degree Field extension over
R
with basis
{
{
1
;
i
;
j
}
=
=−
holding the properties as in
C
. Thus, it is not possible to create the Theory
of Triplets while satisfying the requirements of a Field. Hamilton had to abandon
1
;
i
;
j
}
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