Quaternions:
We will now delve into a subject, which at first may seem quite unrelated.
But
keep the above expression in mind, since it will reappear in most surprising way. This story
begins in the early 19th century, when the great mathematician William Rowan Hamilton
was searching for a generalization of the complex number system.
Imaginary numbers can be thought of as linear combinations of two basis elements, 1 and
i
,
which satisfy the multiplication rules 1
2
= 1,
i
2
=

1 and 1
·
i
=
i
·
1 =
i
. (The interpretation
of
i
=
√

1 arises from the second rule.) A complex number
a
+
bi
can be thought of as a
vector in 2dimensional space (
a, b
). Two important concepts with complex numbers are the
modulus
, which is defined to be
√
a
2
+
b
2
, and the
conjugate
, which is defined to be (
a,

b
). In
vector terms, the modulus is just the length of the vector and the conjugate is just a vertical
reflection about the
x
axis. If a complex number is of modulus 1, then it can be expressed
as (cos
θ,
sin
θ
).
Thus, there is a connection between complex numbers and 2dimensional
rotations. Also, observe that, given such a unit modulus complex number, its conjugate is
(cos
θ,

sin
θ
) = (cos(

θ
)
,
sin(

θ
)). Thus, taking the conjugate is something like negating
the associated angle.
Hamilton was wondering whether this idea could be extended to three dimensional space. You
might reason that, to go from 2D to 3D, you need to replace the single imaginary quantity
i
with two imaginary quantities, say
i
and
j
.
Unfortunately, this this idea does not work.
After many failed attempts, Hamilton finally came up with the idea of, rather than using two
imaginaries, instead using three imaginaries
i
,
j
, and
k
, which behave as follows:
i
2
=
j
2
=
k
2
=
ijk
=

1
ij
=
k, jk
=
i, ki
=
j.
Combining these, it follows that
ji
=

k
,
kj
=

i
and
ik
=

j
.
The skew symmetry of
multiplication (e.g.,
ij
=

ji
) was actually a major leap, since multiplication systems up to
that time had been commutative.)
Hamilton defined a
quaternion
to be a generalized complex number of the form
q
=
q
0
+
q
1
i
+
q
2
j
+
q
3
k.
Thus, a quaternion can be viewed as a 4dimensional vector
q
= (
q
0
, q
1
, q
2
, q
3
).
The first
quantity is a scalar, and the last three define a 3dimensional vector, and so it is a bit more
intuitive to express this as
q
= (
s, u
), where
s
=
q
0
is a scalar and
u
= (
q
1
, q
2
, q
3
) is a vector
in 3space. We can define the same concepts as we did with complex numbers:
Conjugate: q
*
= (
s,

u
)
Modulus:

q

=
p
q
2
0
+
q
2
1
+
q
2
2
+
q
2
3
=
p
s
2
+ (
u
·
u
)
Unit Quaternion: q
is said to be a unit quaternion if

q

= 1
Quaternion Multiplication:
Consider two quaternions
q
= (
s, u
) and
p
= (
t, v
):
q
=
(
s, u
) =
s
+
u
x
i
+
u
y
j
+
u
z
k
p
=
(
t, v
) =
t
+
v
x
i
+
v
y
j
+
v
z
k.
Lecture 6
7
Spring 2018