-----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1 Dear Ian,
I don't see the problem because I did not substitute 'scalar' for 'complex number'. I prefer a more practical approach, so when it is helpful to describe a problem by using the vector space (C, +), I shall do so, and in this context I shall call the complex number vectors. When I want to describe a problem with the field (C, +, *), a shall do so, and I gratefully use functions as exp, log, etc, for which obviously the multiplication is a required operation. Cheers, Tim On 04/02/2014 02:39 PM, Ian Tickle wrote: > On 2 April 2014 12:49, Tim Gruene <t...@shelx.uni-ac.gwdg.de> wrote: > >> P.S.: complex numbers together with the operation '+' defined in >> the canonical way fulfill the axioms of a vector space, hence >> complex number are vectors. If you also take the operation '*' >> into account, defined in the canonical way, (C, +, *) fulfills >> the axioms of a field, which is an algebra, but not a vector >> space. So it depends on your point of you, just like with photons >> and their interpretation as waves or as particles: pick what >> suits you best depending on what you want to describe. >> > > The problem with this is that if you substitute 'scalar' for > 'complex number' the first part of your first sentence becomes > "scalars together with the operation '+' defined in the canonical > way fulfill the axioms of a vector space" which is still a > perfectly true statement. However then applying your conclusion, > i.e. "hence scalars are vectors" we get a false statement because > scalars are not the same thing as vectors in 1-space. A scalar > satisfies all the axioms of a vector space (associativity, > commutativity, distributivity, identity, inverse, addition etc.), > but that doesn't make it a vector, because a scalar satisfies other > axioms (e.g. multiplication & addition) that a vector doesn't. The > additive property of complex numbers (as well as all the axiomatic > properties of a vector space as applied to them) work perfectly > well if you treat them as scalars, e.g. addition: (a + i b) + (c + > i d) = (a+c) + i (b + d), so why lose out on all the other useful > properties of complex numbers (multiplication, complex conjugate > etc) when it's completely unnecessary? > > A complex number is invariably defined as the scalar (a + i b) not > the vector (a,b) because otherwise you miss out on most of its > useful properties that are not shared by vectors. Conversely you > don't lose anything by not treating a complex number as a vector > (e.g. as indicated above vector addition still works). Complex > numbers behave like scalars in all other respects, e.g. we can take > sine, cosine, log, exp etc (and of course multiplication and > division as you point out) of a complex number which you can't do > with a vector. You could of course define say the exp of a complex > number differently as a vector exp(a,b) = (exp(a),exp(b)) (i.e. > like the array function in F90), but again an array is not the > same thing as a vector and also since we have already defined a > complex number as a scalar via de Moivre's theorem: exp(a+ib) = > cos(a) + i sin(b), to define it differently would be very > confusing! > > -- Ian > - -- - -- Dr Tim Gruene Institut fuer anorganische Chemie Tammannstr. 4 D-37077 Goettingen GPG Key ID = A46BEE1A -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.4.12 (GNU/Linux) Comment: Using GnuPG with Icedove - http://www.enigmail.net/ iD8DBQFTPAnrUxlJ7aRr7hoRAo7HAJ0f3ZXJDTEtSwWT+lfagSUo0COQoACfRLlc 5SsyOz89ywCpq/FmikF/qHY= =lTtx -----END PGP SIGNATURE-----
