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THE HYPERCOMPLEXITY
There is a number i, defined by i2=-1, as we saw
in the last section. Why is there ‘a’ number? Why
not
‘two’ such numbers? This idea came to a big man named
Sir William Rowan Hamilton. He took two numbers which
could represent the square root of -1 and reached a
strange conclusion. Let’s see how.
Suppose there are
two numbers, i and j, which are the square roots of
-1. So, i2=j2=-1. But,
this does not mean that i=±j
(as we are not considering them real or obvious
numbers). Now, i2+1=0 and j2+1=0.
Multiply them: first by second and then, second by
first.We get,
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We get,
j2 + i2 + j2 + 1 = j2i2 + j2 + i2 + 1 = 0.
Thus, assuming i2 + j2 = j2 + i2, we get,
i2j2 = j2i2
Let ij=x ji. So, i2j2 = iijj = x4jjii
= x4j2i2 |
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This implies x4 = 1 or, x = ±1
(as it also reflects something new (anti commutativity)
and does not include complex roots).
So, ij = ±ji.
Post multiply the last equation by ij. So, (ij)2 = ±jiij. Now, for RHS, mid term ii = i2 = -1. So,
(ij)2 = -(±jj)
= -(±(-1))
= ±1.
So, we get a new number ij (called k) such that i2 = j2 = ±k2 =-1. |
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From
this, we get, jk =
±kj; ik =
±ki
in the same way as we had obtained ij =
±ji.
Now,
consider (ijk)2 = (ijk)(ijk). We get (ijk)2
=
±1.
If we allow all commutation relations to anti-commute,
we get,
(ijk)2 = -1 (if k2=+1)
= +1 (if k2=-1)
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Consider first (ijk)2 = -1. Then, it is another
complex number. Then, k=±1.
Let p = ijk. Then, ijp = k =±1.
Again, let’s choose negative sign. If we consider the
second, then, p = ijk = -1, but then, k is a complex
number. So, if p is complex, k is -1 and if k is complex,
p is -1. So, we have four quantities, of which three are
complex.
So, i2 = j2
= k2 = ijk =-1. |
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Hamilton
got this idea during a morning walk on ‘Brougham Bridge’
with his wife. His wife might have not liked that walk too
much. But, Hamilton was overjoyed with the discovery. He
inscribed the formula on a stone nearby. Now, what would
have happened if we had chosen some commutation relations
to commute and anti-commute? Then, we would have had a
dirty piece of Mathematics. Anti-commutation relation has
yielded a beautiful equation of complex beauty. A complex
number with i, j, k and 1 as bases is called a quaternion.
So,
q = t + ui + vj + wk.
This is an extended form of complex number. So, you need a
4-D complex plane to represent it. Complex conjugate is
defined in similar way:
p = t – ui – vj – wk.
We have pq = t2 + u2 + v2 +
w2. |
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These
quaternions have fundamental use in rotation. The
transformation q®qi
gives a new quaternion which, in complex plane, represents
a new point q¢,
but at same distance from the origin as q (qp = q¢p¢
= t2 + u2 + v2 + w2).
Similarly, other transformations q®qk
or q®qj
give rotation. |
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Quaternion is something related to 4-D space. But, what about
a general n-dimensional space? A mathematician named
William Kingdon Clifford developed something called
Clifford algebra (both have the same first name William,
which might be coincidence). The components of this
algebra are n quantities which define a reflection. The
quantities are
g1,
g2,
g3, ……
gn (n is the number of dimensions).
gr
causes reversal of rth coordinate (reflection
through (n-1) hyper planes, which do not contain rth
axis). These quantities are defined by:
gi2 = -1;
gigj
= –gjgI (i≠j),
generalization of quanternions. |
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