Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.1 exptools 1/6/84; site hlexa.UUCP Path: utzoo!watmath!clyde!burl!ulysses!mhuxl!houxm!ihnp4!hlexa!phil 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