Real Cayley-Dickson algebras are a class of 2ⁿ-dimensional real algebras containing the real numbers, complex numbers, quaternions, and the octonions (Cayley numbers) as special cases. Each real Cayley-Dickson algebra of dimension greater than eight (a higher dimensional real Cayley-Dickson algebra) is a real normed algebra containing a multiplicative identity and an inverse for each nonzero element. In addition, each element a in the algebra has defined for it a conjugate element ā analogous to the conjugate in the complex numbers. These algebras are not alternative, but are flexible and satisfy the noncommutative Jordan identity. Each element in these algebras can be written A= a₁+ea₂ where e is a basis element and a₁,a₂ are elements of the Cayley-Dickson algebra of next lower dimension. Results include the facts that for each real Cayley-Dickson algebra aⁱaʲ = aⁱ⁺ʲ and (aⁱb)aʲ = aⁱ(baʲ) for all integers i,j and any a,b in the algebra. The major result concerns zero divisors.... --Abstract, page ii
