So what exactly is a complex number?

[Lawrence D'Oliveiro]

In message <roy-4D87D2.00061301092007 at news.panix.com>, Roy Smith wrote:

> In article <fbamkq$r7q$5 at lust.ihug.co.nz>,
>  Lawrence D'Oliveiro <ldo at geek-central.gen.new_zealand> wrote:
> 
>> In message <46d89ba9$0$30380$9b4e6d93 at newsspool4.arcor-online.net>,
>> Wildemar Wildenburger wrote:
>> 
>> > Tim Daneliuk wrote:
>> >> 
>> >> One of the most common uses for Complex Numbers is in what are
>> >> called "vectors".  In a vector, you have both an amount and
>> >> a *direction*.  For example, I can say, "I threw 23 apples in the air
>> >> at a 45 degree angle".  Complex Numbers let us encode both
>> >> the magnitude (23) and the direction (45 degrees) as a "number".
>> >> 
>> > 1. Thats the most creative use for complex numbers I've ever seen. Or
>> > put differently: That's not what you would normally use complex numbers
>> > for.
>> 
>> But that's how they're used in AC circuit theory, as a common example.
> 
> But, if I talk about complex impedance in an AC circuit, I'm measuring two
> fundamentally different things; resistance and reactance.  One of these
> consumes power, the other doesn't.  There is a real, physical, difference
> between these two things. When I talk about having a pole in the 
> left-hand plane, it's critical that I'm talking about negative values for
> the real component.  I can't just pick a different set of axis for my
> complex plane and expect things to still make sense.

In other words, there is a preferred coordinate system for the vectors. Why
does that make them any the less vectors?