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The set of hypercomplex numbers is defined as
 
(1) 
Hypercomplex numbers
define a n+1dimensional complex space with i_{l} orthonormal to i_{m}, for l
m. For all cases presented in this
paper l will be limited to 3. Such numbers are quaternions, which
can be represented as
q=q_{0}+iq_{1}+jq_{2}+kq_{3}, where i,j,k are imaginary numbers that
satisfying the following relations:
 
(2) 
Table 1:
The multiplication table for quaternion algebra.

1 
i 
j 
k 
1 
1 
i 
j 
k 
i 
i 
1 
k 
j 
j 
j 
k 
1 
i 
k 
k 
j 
i 
1 
The multiplication table for quaternion unit vectors is shown in
Table 1.
With these definitions, quaternionic addition between two quaternions,
q and p,
can be defined as
 

 (3) 
and multiplication as
 

 
 
 
 (4) 
Notice that multiplication in equation 4, is not
commutative due to the quaternionic algebra rules defined in table
1.
Quaternions are often separated into two parts, q_{0} and
, respectively called the scalar and vector
part of the quaternion.
Using this definition, the conjugate of q, , is
 
(5) 
and the norm of a quaternion is defined by
 
(6) 
It is useful to formulate a polar representation of the
quaternion, as this will be the primary notation throughout this
paper. For any complex number, , the argument or
phaseangle is defined as atan2(b,a). If is written it
the form , then is the
phase (argument) of , denoted arg()=.Quaternions contain three complex subfields and, correspondingly, three
phases that are the projections onto the i, j, or kcomplex plane,
 
(7) 
 (8) 
 (9) 