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 <[log in to unmask]> 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 <[log in to unmask]>
> > > Sent by: CCP4 bulletin board <[log in to unmask]>
> > > 10/13/2010 01:48 PM
> > > Please respond to
> > > William Scott <[log in to unmask]>
> > >
> > >
> > > To
> > > [log in to unmask]
> > > 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 /