Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.1 6/24/83; site watmath.UUCP Path: utzoo!watmath!csc From: csc@watmath.UUCP (Computer Sci Club) Newsgroups: net.math Subject: N dimentional numbers? Message-ID: <6966@watmath.UUCP> Date: Tue, 21-Feb-84 13:11:11 EST Article-I.D.: watmath.6966 Posted: Tue Feb 21 13:11:11 1984 Date-Received: Wed, 22-Feb-84 02:00:26 EST Organization: U of Waterloo, Ontario Lines: 32 The question was asked are there N-dimensional numbers in the same way that the complex numbers are 2-dimensional. The answer depends on what you mean by "numbers". If you mean fields then the answer is no. A field F is a finite extention of the reals R if there exist a1,a2, ... aN elements of F such that any element f of F can be written as f= r1*a1 + r2*a2 + ... + rN*aN with r1, ... rN elements of R. The only finite extention of the reals is C the complex numbers. Sketch of proof (see Herstein, Topics in Algebra, chapter 7, (read chapters 1-6 first)) If F is a finite extention of R then any element f of F is a root of a polynomial in R. (basic result of field theory). All roots of polynomials in R are complex numbers. Therefore f is a complex number, therefore F is contained in C (and can be shown equal to C). There are field extentions of R (fields which contain a copy of R), for example the field of rational functions p(x)/q(x) where p and q are real polynomials and q is non zero. But these extentions are infinite dimensional. If we relax our definitions of numbers to include division rings (like fields but ab=ba is not necessarily true) then we get one more finite extention of the reals the real quaternions. (They look like a + b*i + c*j + d*k and are four dimensional over the reals) (This is the only algebraic division ring extention, I am not sure if it is the only finite division ring extention) William Hughes