
This chapter is not written for mathematicians, but to give a very idea about complex numbers.
What is the squareroot of 1 (sqrt(1))? This question leads into the theory of complex numbers.
The solutions of the equitation x²  4 = 0 are 2 and 2. Generally speaking the equitation x²  s = 0 is solved by
sqrt(s) and sqrt(s) if s is a positive number.
In case of s = 0 there is only one solution: x = 0.
What happens if s is a negative number? How do you solve x² + 1 = 0? In this case we got: x = +/sqrt(1). For this problem we have no solution yet. So we define sqrt(1) = i and call i the Imaginary Unit. By this definition any equitation x²  s = 0 can be solved for any s. For negative s the solutions are x = +/ sqrt(s)*i. So for any negative s we can use i = sqrt(1) scaled by sqrt(s) which is a normal (real) number.
For you don't know wheather s is positive or negative, we define a complex number as a pair of normal (real) numbers (a,b). If s is positive, the solutions are (+/sqrt(s),0), else (0,+/sqrt(s)). There are also complex numbers with a normal and a imaginary part, i.e. (3,7). These can also be written as a + b*i, i.e. 3 + 7*i. For you can understand a complex number as a pair of normal numbers, the pair (a,b) are coordinates. You can draw a complex number in a plain of coordinates. a gives you the xcoordinate and b the ycoordinate.
How to calculate complex numbers
You add two complx numbers by adding their normal parts (real part) and adding their imaginary parts, like this:
(3,4) + (5,2) = (8,6) or (3 + 4*i) + (5 + 2*i) = (3 + 5) + (4 + 2)*i = 8 + 6*i
The multiplication of two complex numbers is more difficult:
(3,4) * (5,2) = (7,26)
But this is not too strange, if you multiply two complex numbers with each other like this:
(i * i = 1 (!))
(3 + 4*i) * (5 + 2*i) = 15 + 6*i + 20*i + 8*i*i = 15  8 + 26*i = 7 + 26*i


