The definition game is on! :) Vectors are supposed to have direction and amplitude, unlike scalars. 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 /
