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

Speaker: **Warren D. Smith**, DIMACS, Rutgers University, and Temple
University Mathematics Department

Date: Friday September 13, 2002, 1:10 pm

Location: DIMACS Seminar Room, CoRE Building, Room 431A, Rutgers University.

Abstract:

The ``Cayley-Dickson process,'' 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 close the book, state that these are the full set of division (or normed) algebras with $1$ over the real numbers. Their properties keep degrading: the reals are ordered and self-conjugate, but the complex numbers lose these properties; at the quaternions we lose commutativity; and at the octonions we lose associativity. If one keeps Cayley-Dickson doubling to get the 16D ``sedenions,'' zero-divisors appear.

We introduce a different doubling process which also produces the
complexes, quaternions, and octonions, but keeps going to yield
$2^n$-dimensional normed algebraic structures, *with division*,
for every $n > 0$. Each contains all the previous ones as
subalgebras. We'll see how these evade the Famous
Impossibility Theorems.

But properties continue to degrade. The 16-ons lose
distributivity. The 32-ons lose the property that the solutions of
generic division problems are *unique*.

All the $2^n$-ons have $1$ and obey numerous identities including
weakened distributive and associative 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 $2^n$-ons are
also unique, albeit in a much weaker sense. All the $2^n$-ons support
a version of the fundamental theorem of algebra. Normed algebras
(rational but not nec. distributive) over the reals and complex
numbers are impossible in dimensions other than powers of 2.