TITLE
Quaternions, octonions, and now, 16-ons and $2^n$-ons;
New kinds of numbers.
AUTHOR
Warren D. Smith
ABSTRACT
``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 ($\overline{\overline{x}} = x$)
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 \emph{different} doubling process
which also produces the complexes, quaternions, and octonions, but
keeps going to yield $2^n$-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 $2^k-1$ dimensions
for each $k \ge 2$; this operation is impossible
in other dimensions.
But properties continue to degrade. The 16-ons
lose distributivity,
right-cancellation $yx \cdot x^{-1} = y$,
flexibility $a \cdot ba = ab \cdot a$, and
antiautomorphism $\overline{ab} = \overline{b} \overline{a}$.
The 32-ons 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 $2^n$-ons
are generalized smooth for $n \le 4$.
All the $2^n$-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
\emph{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.
The $2^n$-ons with $n \le 4$ 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.
KEYWORDS
Quaternions, octonions, 16-ons,
fundamental theorem of algebra, division,
loops, non-distributive algebras, topology,
generalized smoothness,
Brouwer degree, vector fields on spheres,
vector product, weak-linearity, left-alternative,
Moufang and Bol laws, Schwartz-Zippel lemma,
automatic verification of polynomial identities,
Trotter product limit formula.