So what exactly is a complex number?

[Roy Smith]

Wildemar Wildenburger <lasses_weil at klapptsowieso.net> wrote:
> Again, that is just one way to interpret them. Complex numbers are not 
> vectors (at least no moe than real numbers are).

OK, let me take a shot at this.

Math folks like to group numbers into sets.  One of the most common sets is 
the set of integers.  I'm not sure what the formal definition of an integer 
is, but I expect you know what they are: 0, 1, 2, 3, 4, etc, plus the 
negative versions of these: -1, -2, -3, etc.

The set of integers have a few interesting properties.  For example, any 
integer plus any other integer gives you another integer.  Math geeks would 
say that as, "The set of integers is closed under addition".

Likewise for subtraction; any integer subtracted from any other integer 
gives you another integers.  Thus, the set of integers is closed under 
subtraction as well.  And multiplication.  But, division is a bit funky.  
Some integers divided by some integers give you integers (i.e. 6 / 2 = 3), 
but some done (i.e. 5 / 2 = 2.5).

So, now we need another kind of number, which we call reals (please, no nit 
picking about rationals).  Reals are cool.  Not only are the closed under 
addition, subtraction, and multiplication, but division too.  Any real 
number divided by any other real number gives another real number.

But, it's not closed over *every* possible operation.  For example, square 
root.  If you take the square root of 4.23, you get some real number.  But, 
if you try to take the square root of a negative number, you can't do it.  
There is no real number which, when you square it, gives you (to use the 
cannonical example), -1.  That's where imaginary numbers come in.  The math 
geeks invented a wonderful magic number called i (or sometimes j), which 
gives you -1 when you square it.

So, the next step is to take an imaginary number and add it to a real 
number.  Now you've got a complex number.  There's all kinds of wonderful 
things you can do with complex numbers, but this posting is long enough 
already :-)