Introduction:
First and foremost, complex numbers use the symbol i. When dealing with complex numbers, $ i = sqrt(-1) $. Since there is no Real representation for this, the square root of -1 is represented as the lower-case, italic "i". In engineering, the imaginary number is also often denoted with a j in place of the i.
Powers of i:
The following properties follow from the definition of an imaginary:
i0 = 1
i1 = $ \sqrt(-1) $
i2 = -1
i3 = -i
--pattern repeats--
i4 = 1
i5 = $ \sqrt(-1) $
.
.
.
in
This pattern repeats for i^n. Therefore, if i^n, then n modulo 4 (the remainder of n/4) yields its place in the repeating pattern.
Conjugation:
The second most important part of complex numbers (in the author's opinion) is the idea of conjugation. This is key in the actual manipulation of complex numbers in equations and general mathematical use.
The use of conjugates is to remove the imaginary component (the i's) from a part of an equation (if not all of it).
A complex number multiplied by its conjugate will yield a real result.
Example:
$ (a - b*i) $ has conjugate $ (a + b*i) $ and $ (a - b*i)*(a + b*i) = a^2 - b^2 $
$ (f*i + g) $ has conjugate $ (-f*i + g) $ and $ (f*i + g)*(-f*i + g) = -f^2 + g^2 $
