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From: phil@hlexa.UUCP (Phil Fleming)
Newsgroups: net.math
Subject: Re: N-dimensional numbers?
Message-ID: <1346@hlexa.UUCP>
Date: Thu, 23-Feb-84 09:59:38 EST
Article-I.D.: hlexa.1346
Posted: Thu Feb 23 09:59:38 1984
Date-Received: Fri, 24-Feb-84 01:00:30 EST
Organization: AT&T Bell Laboratories, Short Hills, NJ
Lines: 39

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, NJ