POSSIBLE PREFACE (written in LATEX) for book
"Quaternions, octonions, and now, 16-ons and $2^n$-ons;
New kinds of numbers"
by Warren D. Smith February 2004
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This book introduces a new kind of numbers called ``$2^n$-ons.''
The previously known real numbers, complex numbers,
quaternions, and octonions are the 1, 2, 4, and 8-dimensional
special cases of the $2^n$-ons, i.e., the
1-ons, 2-ons, 4-ons, and 8-ons, respectively.
By means of a new dimension-doubling process, we show how
to extend these into dimensions 16, 32, and so on forever.
Why are only powers of 2 permitted as dimensions?
One of the most important properties of a $2^n$-on
$x$ is that it has a norm $|x|$
(corresponding to its \emph{length} as a Euclidean vector)
such that if $ab = c$, then $|a| \cdot |b| = |c|$.
We shall see that this norm-multiplicativity
requirement, combined with the demand that the multiplication
law be computable solely by using the 4 arithmetic operations,
prevents any dimensions besides powers of 2.
If we restrict ourselves to the \emph{three}
non-divisive arithmetic operations $\{+ , - , \cdot \}$
then famous theorems show that
only dimensions 1,2,4,8 are permissible;
our point is that this restriction is artificial and
there is no need to stop at the 8-ons.
Why should people care about the $2^n$-ons, and
why have they cared about quaternions and octonions?
The top reason is that among all multidimensional
algebraic entities, these are the
ones most deserving to be called ``numbers:''
\begin{enumerate}
\item
Numbers may be added, subtracted, or multiplied to get another.
\item
There are a continuum infinity of numbers.
\item
There is a unique number called $1$ such that $1x=x1=x$.
\item
Another unique number, called $0$, is such that $0x=x0=0$ and $x+0=0+x=x$.
\item
$x(y+z) = xy + xz$.
\item
If two numbers are multiplied, their lengths multiply.
The length of a nonzero number is a positive real.
\item
If $x \ne 0$, then there is a unique number $x^{-1}$ such that
$x x^{-1} = x^{-1} x = 1$.
\item
If $x y = z$, then $y = x^{-1} z$.
\item
Exponentiation to integer powers obeys $x^p x^q = x^{p+q}$
and $(x^p)^q = x^{pq}$.
\end{enumerate}
A second reason is that the $2^n$-ons generalize the
extremely useful 3D notion of the \emph{vector product}
$\vec{a} = \vec{b} \times \vec{c}$
(beloved by physicists)
to all dimensions of the form $2^r - 1$.
(Again, we shall show that these are the only
permissible dimensions.)
A third reason is that
complex numbers, quaternions and octonions have been useful
for understanding, and for computing in, low-dimensional geometry.
For example, the quaternions yield a superior way to represent
3D and 4D rotations. Will the $2^n$-ons be equally useful?
Finally, the fourth reason is that one would like to be
able to \emph{divide} two numbers, that is, to solve
$w x = y$ or $x w = y$ for $x$. We shall show that
solutions to generic division problems exist and
are unique in the 16-ons, again surprisingly evading a famous
impossibility theorem.
The $2^n$-ons obey and disobey a whole world of properties,
and much of the book is devoted to exploring them and
to investigating ways in which our $2^n$-on definition
is ``unique'' and best possible.''
This book also contains three new contributions of
non-$2^n$-onic interest:
\begin{enumerate}
\item
New ways are invented to make your computer prove identities,
instead of you. This should be of interest to anybody
who cares about computer algebra.
\item
We introduce a new notion called ``generalized smoothness''
into topology which enables many important topological results
to be applied even in the \emph{absence} of continuity.
The 16-on multiplication map is generalized smooth.
But anybody interested in continuum topology would be
strongly advised to comprehend this notion because it
probably has many applications that have nothing to do
with 16-ons.
\item
We introduce for the first time a ``taxonomy of loops,''
organizing loop theory into phylla and species
to bring order where there once was chaos.
(``Loops'' are non-associative groups.)
This taxonomy suits our purposes by providing a ready-made
pre-organized set of valid and invalid $2^n$-on identities.
But again, it should be of wider interest.
Indeed, anyone entering the wilds of loop theory without
first reading this taxonomy, will be in the position
of a novice entering a new country without a map.
\end{enumerate}
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