Graphics Reference
In-Depth Information
Appendix C
The Four
n
-Square Algebras
C.1 Introduction
The magnitude of a real quantity is its positive value. However, when dealing with
objects such as complex numbers, quaternions and octonions, their magnitude is
expressed using the Pythagorean formula which takes the square root of the sums of
the terms squared.
For example, the magnitude of a complex number
z
1
=
a
+
bi
is
a
2
b
2
|
z
1
|=
+
and the magnitude of a quaternion
q
=
s
+
x
i
+
y
j
+
z
k
is
s
2
x
2
y
2
z
2
|
q
|=
+
+
+
and something similar for an octonion, which has 8 terms.
In their topic
On Quaternions and Octonions
[
9
], John Conway and Derek Smith
use the Euclidean norm
N
to represe
nt
the sums of the squares, although other au-
thors define the Euclidean norm as
√
N
. However, for the purpose of this description
I will employ Conway and Smith's definition. Thus
a
2
b
2
N(a
+
bi)
=
+
and
s
2
x
2
y
2
z
2
.
=
+
+
+
N(
q
)
We know from the algebra of complex numbers that
|
z
1
||
z
2
|=|
z
1
z
2
|
or
N(z
1
)N(z
2
)
=
N(z
1
z
2
)
and from the algebra of quaternions that
|
q
1
||
q
2
|=|
q
1
q
2
|
