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Basic Algebra:
         An octonion is a hypercomplex number which can be written as a linear combination of eight basal elements.   A set of octonions forms an algebra.  If A, B, C represent three octonions, the algebra of octonions, denoted by O is
  • Non commutative :   AB is not necessarily equal to BA
  • Non associative    :   (AB)C is not necessarily equal to A(BC)
  • Alternative            :   A(AB) = A2 B
  • Power associative :   An Am = An+m
And, octonions form a
  • Composition algebra :  n(AB) = n(A) n(B) ,
          where the norm of A , denoted n(A), is defined as
          n(A) = AA* = A*A , A* is a conjugate of A
    Division algebra :   For a not zero octonion, A, the multiplicative inverse A-1 is also an octonion.
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