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

“Cayley-Dickson doubling,” starting from the real numbers, successively yields the complex numbers (dimension 2), quaternions (4), and octonions (8). Each contains all the previous ones as subalgebras. Famous Theorems, previously 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 complex numbers lose the ordering and self-conjugacy (x = x) properties of the reals; at the quaternions we lose commutativity; 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 different doubling process which also produces the complexes, quaternions, and octonions, but keeps going to yield 2-dimensional normed algebraic structures, for every n > 0. Each contains all the previous ones as subalgebras. We’ll see how these evade the Famous Impossibility Theorems. They also lead to a rational “vector product” operation in 2 − 1 dimensions for each k ≥ 2; this operation is impossible in other dimensions. But properties continue to degrade. The 16-ons lose distributivity, right-cancellation yx · x = y, flexibility a · ba = ab · a, and antiautomorphism ab = ba. The 32ons lose the properties that the solutions of generic division 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 2-ons are generalized smooth for n ≤ 4. All the 2-ons have 1 and obey numerous identities including weakenings of the distributive, associative, and antiautomorphism laws. In the case of 16-ons these weakened distributivity laws characterize them, i.e. our 16-ons are, in a sense, unique and best-possible. Our 2-ons are also unique, albeit in a much weaker sense. The 2-ons with n ≤ 4 support a version of the fundamental theorem of algebra. Normed algebras (rational but not nec. distributive) over the reals are impossible in dimensions other than powers of 2.

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