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
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