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Alternate Arithmetics on the Numbers a + bi
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ab)b absolute value addition ai*i algebra of Cayley algebra of complex algebra of quaternions algebra of real anin arbitrary real numbers assertion 19.2 associative algebras associative division algebra bases basis vectors called Cayley numbers chapter coefficients coincides commutative division algebra complex numbers conjugation consider coordinates defined definition denote determined division system equality equation example expressions fact follows formula Frobenius theorem given Hence holds Hurwitz's theorem hypercomplex numbers hypercomplex system iaip identity imaginary units implies initial basis isomorphic lemma linear combination linearly independent mathematician matrices of order means multiplication rule multiplication table n-dimensional algebra n-dimensional vector nilpotent ideal nonzero vector normed algebra number system obtain operation orthogonal transformation orthonormal basis perpendicular problem properties quaternion algebra representation result rotation satisfies scalar product semisimple solution subalgebra subspace sum of squares summand system of numbers system of vectors unit vector vector space vectors 91 xiyi
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