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Quaternion
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The domain constructor Quaternion implements quaternions over
commutative rings.

The basic operation for creating quaternions is quatern.  This is a
quaternion over the rational numbers. ::

  q := quatern(2/11,-8,3/4,1)
     2        3
    -- - 8i + - j + k
    11        4
                        Type: Quaternion Fraction Integer

The four arguments are the real part, the i imaginary part, the j
imaginary part, and the k imaginary part, respectively. ::

  [real q, imagI q, imagJ q, imagK q]
      2     3
    [--,- 8,-,1]
     11     4
                        Type: List Fraction Integer

Because q is over the rationals (and nonzero), you can invert it. ::

  inv q
      352     15488      484       1936
    ------ + ------ i - ----- j - ------ k
    126993   126993     42331     126993
                        Type: Quaternion Fraction Integer

The usual arithmetic (ring) operations are available ::

  q^6
      2029490709319345   48251690851     144755072553     48251690851
    - ---------------- - ----------- i + ------------ j + ----------- k
         7256313856        1288408         41229056         10307264
                        Type: Quaternion Fraction Integer

  r := quatern(-2,3,23/9,-89); q + r
      20        119
    - -- - 5i + --- j - 88k
      11         36
                        Type: Quaternion Fraction Integer

In general, multiplication is not commutative. ::

  q * r - r * q
      2495             817
    - ---- i - 1418j - --- k
       18               18
                         Type: Quaternion Fraction Integer

There are no predefined constants for the imaginary i, j, and k parts,
but you can easily define them. ::

  i:=quatern(0,1,0,0) 
    i
                         Type: Quaternion Integer

  j:=quatern(0,0,1,0) 
    j
                         Type: Quaternion Integer
 
  k:=quatern(0,0,0,1)
    k
                         Type: Quaternion Integer

These satisfy the normal identities. ::

  [i*i, j*j, k*k, i*j, j*k, k*i, q*i]
                             2         3
     [- 1,- 1,- 1,k,i,j,8 + -- i + j - - k]
                            11         4
                          Type: List Quaternion Fraction Integer

The norm is the quaternion times its conjugate. ::

  norm q
    126993
    ------
     1936
                          Type: Fraction Integer

  conjugate q 
      2        3
     -- + 8i - - j - k
     11        4
                          Type: Quaternion Fraction Integer

  q * %
     126993
     ------
      1936
                          Type: Quaternion Fraction Integer

See Also:

* )help Octonion
* )help Complex
* )help CliffordAlgebra
* )show Quaternion