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 ================================================================= 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} ========================================================================