Elementary Derivations of the Real Composition Algebras

"Real Normed Algebras Revisited", the last paper of the late Gadi Moran, attempts to reconstruct the discovery of the complex numbers, the quaternions and the octonions, as well as proofs of their properties, using only what was known to 19th century mathematicians. In his research, Gadi had discovered simple and elegant proofs of the above mentioned classical results using only basic properties of the geometry of Euclidean spaces and tools from high school geometry. Although his reconstructions are different from the original derivations by Hamilton and others [van der Waerden 1976], they underline an interesting connection between Euclidean geometry and these algebras. The goal of this manuscript is to present Gadi's derivations in a way which is accessible to a wide audience of math readers.