Quaternions, octonions, and now, 16-ons and 2n-ons; New kinds of numbers.

Abstract | \Cayley-Dickson doubling, " starting from the real num-bers, successively yields the complex numbers (dimension 2), quaternions (4), and octonions (8). Each contains all the previous ones as subalgebras. Famous Theorems, pre-viously thought to be the last word, state that these are the full set of division (or normed) algebras with 1 over the real numbers. Their properties keep degrading: the com-plex numbers lose the ordering and self-conjugacy (x = x) properties of the reals; at the quaternions we lose commu-tativity; and at the octonions we lose associativity. If one keeps Cayley-Dickson doubling to get the 16-dimensional \sedenions, " zero-divisors appear. We introduce a dierent doubling process which also pro-duces the complexes, quaternions, and octonions, but keeps going to yield 2n-dimensional normed algebraic structures, for every n> 0. Each contains all the pre-vious ones as subalgebras. We'll see how these evade the Famous Impossibility Theorems. They also lead to a ra-tional \vector product " operation in 2k 1 dimensions for each k 2; this operation is impossible in other dimen-sions. But properties continue to degrade. The 16-ons lose distributivity, right-cancellation yx x1 = y, exibility a ba = ab a, and antiautomorphism ab = ba. The 32-ons lose the properties that the solutions of generic divi-sion problems necessarily exist and are unique, and they lose the \Trotter product limit formula. " We introduce an important new notion to topology we call \generalized smoothness. " The 2n-ons are generalized smooth for n 4. All the 2n-ons have 1 and obey numerous identities in-cluding weakenings of the distributive, associative, and antiautomorphism laws. In the case of 16-ons these weak-ened distributivity laws characterize them, i.e. our 16-ons are, in a sense, unique and best-possible. Our 2n-ons are also unique, albeit in a much weaker sense. The 2n-ons with n 4 support a version of the fundamental theo-rem of algebra. Normed algebras (rational but not nec. distributive) over the reals are impossible in dimensions other than powers of 2