phil@hlexa.UUCP (Phil Fleming) (02/23/84)
J.F. Adams, an algebraic topologist, proved (around 1962) that if
n-dimensional Euclidean space has an operation (henceforth called *)
satisfying;
(1) continuity in the arguments
(2) if x*y=0 then x=0 or y=0
then n=1,2,4 or 8. These operations can be realized by
n=1 Ordinary real number multiplication.
n=2 Pairs of real numbers (called complex numbers)
with (a,b)*(c,d)=(ac - bd, ad + bc).
n=4 Pairs of complex numbers (called quaternions)
with (a,b)*(c,d)=(ac - b(d_), ad + b(c_) ).
Here, _ denotes complex conjugation.
n=8 Pairs of quaternions (called Cayley numbers)
with (a,b)*(c,d)=( ac - b(d_), ad + b(c_) ).
Here, _ denotes quaternionic conjugation
(a,b)_ = ( a_ , -b ) (where a,b are complex
numbers.
You could of course continue this process to get an
operation on 16-dimensional space but it would not
satisfy property (2). The importance of prop(2) is that
it makes division a well defined operation. As a final remark
note that the quaternions are not commutative and the Cayley
numbers are not associative ( (xy)z != x(yz) ).
Reference: J.F. Adams, Vector-fields on Speres, Bull.AMS,68:39-41 (1962).
Phil Fleming
AT&T Bell Laboratories
Short Hills, NJsonnens@mprvaxa.UUCP (Dan Sonnenschein) (02/23/84)
There are also "8-dimensional" numbers called Cayley numbers, after the prominent 19th-century mathematician Arthur Cayley. This number system is non-associative as well as non-commutative. Dan Sonnenschein, ...uw-beaver!ubc-vision!mprvaxa!sonnens
liberte@uiucdcs.UUCP (liberte ) (03/06/84)
#R:hlexa:-134600:uiucdcs:28200032:000:266 uiucdcs!liberte Mar 5 22:38:00 1984 A generalized extension to complex numbers has been done by Charles Muses. I forget how far he has gone with it, but 8 levels sounds about right. So 16, 32 and 64 dimensional numbers are conceivable. At each hier level, more properties drop out. Daniel LaLiberte