So what exactly is a complex number?

[Bjoern Schliessmann]

Grzegorz S?odkowicz wrote:
> 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. 

Indeed, interesting.

> 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?

I didn't say that you must not know the point of application, but I
said that it was not a property of the vector itself. It is true,
however, that in physical calculations you should not "mix" many
types of vectors (like force) that are, in the experiment, applied
to different points of application.

> 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).

Oh, I thought we were talking about quite mathematical vectors? In
physics, I learned that a vector is only what transforms like a
vector.

>> 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.

Indeed, you're right. So, those vectors have different properties
depending on the used coordinate system? I myself prefer the
concise definition ...

Regards,


Björn

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