So what exactly is a complex number?

[Roy Smith]

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.

Well, not really.  They're often talked about as vectors, when people are 
being sloppy, but they really aren't.

In the physical world, let's say I take out a compass, mark off a bearing 
of 045 (north-east), and walk in that direction at a speed of 5 MPH.  
That's a vector.  The "north" and "east" components of the vector are both 
measuring fundamentally identical quantities, along perpendicular axes.  I 
could pick any arbitrary direction to call 0 (magnetic north, true north, 
grid north, or for those into air navigation, the 000 VOR radial) and all 
that happens is I have to rotate my map.

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.