I don't see any conflict here: all you're saying is that there's a 1-to-1 mapping between the complex scalar a+i*b in C and the 2-D vector (a,b) in R^2.
However the vector does not have all the properties of the original complex scalar: for example I can happily compute a value for log(a+i*b) but if I try to compute log((a,b)) the compiler will throw a fit! Of course if you can get along without taking the log, or any other transcendental function, and without doing any algebra using the multiplication, division or power operators, then it won't matter in practice whether you regard it as a+i*b or (a,b). Personally I find it much cleaner to treat a complex number as the scalar a+i*b because then I have the full math library of the Fortran compiler (or whatever is your favourite) at my disposal. Cheers -- Ian On Thu, Oct 14, 2010 at 1:24 PM, Tim Gruene <t...@shelx.uni-ac.gwdg.de> 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) >> > > > -- > -- > Tim Gruene > Institut fuer anorganische Chemie > Tammannstr. 4 > D-37077 Goettingen > > phone: +49 (0)551 39 22149 > > GPG Key ID = A46BEE1A > > > -----BEGIN PGP SIGNATURE----- > Version: GnuPG v1.4.9 (GNU/Linux) > > iD4DBQFMtvZ0UxlJ7aRr7hoRAginAJ4ty3BHIL/h9BXe6U77+/64+cU5UgCY7NEn > F64lkymlpzNz2b2BU3aOFQ== > =dtZf > -----END PGP SIGNATURE----- > >
