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
