So what exactly is a complex number?

[Grzegorz Słodkowicz]

 >> In fact, a proper vector in physics has 4 features: point of
>> application, magnitude, direction and sense.
>>     
>
> No -- a vector has the properties "magnitude" and direction.
> Although not everything that has magnitude and direction is a
> vector.
>
> It's very unusual to have a fixed point of application as a vector's
> property (at least I haven't seen it so far). That would complicate
> equality tests.
>   
Interesting. It appears that we are ran into a mathematical cultural 
difference. Were I come from vectors *are* defined as having four 
properties that I enumerated. After some research I found that English 
sources (Wikipedia) indeed give the definition you supplied. In my old 
mechanics textbook (which is in English) vectors are divided into 
'fixed' (having a clearly defined point of application), 'sliding' 
(p.o.a. can be moved along a line) and 'free' (those not associated with 
a unique line in space).

However, given the following problem: (assuming 2-d Cartesian coordinate 
system and gravity acting 'downwards') "There are 3 point masses: 2 kg 
at (0, 0), 1 kg at (5, 4) and 4 kg at (2, 2). The acting forces are 
given as vectors: [2, 2] [1, 1]. Find the trajectories of all point 
masses." how would you propose to solve it without knowing where the 
forces are applied?
>> In case of a vector in two dimensions (a special case, which you
>> also fail to stress not to mention that you were talking about
>> space) the magnitude and sense can be described by one number     
>
> Actually, the "magnitude" and "sense" you use here are redundant.
> What's the difference between a vector with magnitude "1" and
> sense "-", and magnitude "-1" and sense "+"?
>   
Again, I think we were given different definitions. Mine states that 
direction is 'the line on which the vector lies', sense is the 'arrow' 
and magnitude is the 'length' (thus non-negative). The definition is 
separate from mathematical description (which can be '[1 1] applied at 
(0, 0)' or 'sqrt(2) at 45 deg applied at (0, 0)' or any other that is 
unambiguous).
>> and the direction as another.
>>     
>
> Represent the direction as one number? Only in a one-dimensional
> space.
>   
No. In one-dimensional 'space' direction is a ± quantity (a 'sense'). In 
2-d it can be given as an angle.

Regards,
Greg