So what exactly is a complex number?

[Lawrence D'Oliveiro]

In message <tijpq4-uo22.ln1 at ozzie.tundraware.com>, Tim Daneliuk wrote:

> No, but go to my other example of an aircraft in flight and winds
> aloft.  It is exactly the case that complex numbers provide a convenient
> way to add these two "vectors" (don't wince, please) to provide the
> effective speed and direction of the aircraft.  Numerous such examples
> abound in physics, circuit analysis, the analysis of rotating machinery,
> etc.

Not really. The thing with complex numbers is that they're
numbers--mathematically, they comprise a "number system" with operations
called "addition", "subtraction", "multiplication" and "division" having
certain well-defined properties (e.g. associativity of multiplication, all
numbers except possibly one not having a multiplicative inverse).

An aircraft in flight amidst winds needs only vector addition (and possibly
scalar multiplication) among the basic operations to compute its path--you
don't need to work with complex numbers as such for that purpose.

For my AC circuit theory example, however, you do.