> The definition game is on! :) > > Vectors are supposed to have direction and amplitude, unlike scalars.
I think that this is part of the problem here. Whilst vector quantities do possess both size and direction, not everything that possesses size and direction is necessarily a vector by definition. Thus, just because complex numbers possess an amplitude and a phase angle, that does not automatically make them vectors. The complex numbers are in fact a vector space over the real numbers, but that requires further justification. > Curiously, one can take a position that real numbers are vectors too, if > you consider negative and positive numbers having opposite directions > (and thus subtraction is simply a case of addition of a negative > number). And of course, both scalars and vectors are simply tensors, of > zeroth and first order :) > > Guess my point is that definitions are a matter of choice in math and if > vector is defined as an array which must obey addition and scaling rules > (but there is no fixed multiplication rule - regular 3D vectors have > more than one possible product), then complex numbers are vectors. In a > narrow sense of a real space vectors (the arrow thingy) they are not. > Thus, complex number is a Vector, but not the vector (futile attempt at > using articles by someone organically suffering from article dyslexia). > > Cheers, > > Ed. > > > On Thu, 2010-10-14 at 14:24 +0200, Tim Gruene wrote: >> On Thu, Oct 14, 2010 at 12:34:30PM +0100, Ian Tickle wrote: >> > Formally, a complex number (e.g. a structure factor) is not a vector. >> Formally, C is isomorphous to R^2 (at least that's what math departments >> in >> Germany teach, and it's not difficult to prove), therefore complex >> numbers are >> vectors. That's is unaffected by whether there is a ring-isomorphism >> between C >> and R^2, and it's correct that the elements of a field are usually not >> called >> 'vectors', but that does not mean that it is wrong to consider a complex >> number >> a vector. >> >> Tim >> >> > Just because the addition & subtraction rules (i.e. 'a+b' & 'a-b') are >> > defined for real numbers, complex numbers and vectors doesn't make a >> > complex number a vector, any more than it makes a real number a vector >> > (or vice versa). Entities are defined according to the rules of >> > algebra that they obey, thus real and complex numbers obey the same >> > rules, i.e. the familiar addition, subtraction, multiplication, >> > division & raising to a power. Hence real and complex numbers are >> > both scalars: a real number is a special case of a complex scalar with >> > zero imaginary part (one could program an algorithm for reals using >> > only complex variables & functions and still get the right answer). >> > This also means that the transcendental functions (sin, cos, tan, exp, >> > log etc) are all defined equally well for both real and complex >> > scalars, but not for vectors, a property that programmers in Fortran, >> > C & C++ (and probably others) will be familiar with. Of the addition, >> > subtraction, multiplication, division & power rules, vectors only obey >> > the first two, but unlike real & complex scalars they also obey the >> > scalar product and exterior product rules. >> > >> > The general rule is that "if and only if it looks like a duck, waddles >> > like a duck and quacks like a duck, then it is a duck" - complex >> > numbers might look like vectors but they neither waddle nor quack like >> > them! >> > >> > Cheers >> > >> > -- Ian >> > >> > On Wed, Oct 13, 2010 at 9:57 PM, Yong Y Wang <wang_yon...@lilly.com> >> wrote: >> > > It is already vertical, relative to the real part of Fa (in red), >> i.e. the >> > > blue vector is always vertical to the red vector in this picture >> (and >> > > counter-clockwise). >> > > >> > > Yong >> > > >> > > >> > > >> > > >> > > William Scott <wgsc...@chemistry.ucsc.edu> >> > > Sent by: CCP4 bulletin board <CCP4BB@JISCMAIL.AC.UK> >> > > 10/13/2010 01:48 PM >> > > Please respond to >> > > William Scott <wgsc...@chemistry.ucsc.edu> >> > > >> > > >> > > To >> > > CCP4BB@JISCMAIL.AC.UK >> > > cc >> > > >> > > Subject >> > > [ccp4bb] embarrassingly simple MAD phasing question >> > > >> > > >> > > >> > > >> > > >> > > >> > > Hi Citizens: >> > > >> > > Try not to laugh. >> > > >> > > I have an embarrassingly simple MAD phasing question: >> > > >> > > Why is it that F" in this picture isn't required to be vertical >> (purely >> > > imaginary)? >> > > >> > > http://www.doe-mbi.ucla.edu/~sawaya/tutorials/Phasing/phase.gif >> > > >> > > (Similarly in the Harker diagram of the intersection of phase >> circles, one >> > > sees this.) >> > > >> > > I had a student ask me and I realized that there is this fundamental >> gap >> > > in my understanding. >> > > >> > > Many thanks in advance. >> > > >> > > -- Bill >> > > >> > > >> > > >> > > >> > > William G. Scott >> > > Professor >> > > Department of Chemistry and Biochemistry >> > > and The Center for the Molecular Biology of RNA >> > > 228 Sinsheimer Laboratories >> > > University of California at Santa Cruz >> > > Santa Cruz, California 95064 >> > > USA >> > > >> > > phone: +1-831-459-5367 (office) >> > > +1-831-459-5292 (lab) >> > > fax: +1-831-4593139 (fax) >> > > >> > > -- > Edwin Pozharski, PhD, Assistant Professor > University of Maryland, Baltimore > ---------------------------------------------- > When the Way is forgotten duty and justice appear; > Then knowledge and wisdom are born along with hypocrisy. > When harmonious relationships dissolve then respect and devotion arise; > When a nation falls to chaos then loyalty and patriotism are born. > ------------------------------ / Lao Tse / >
