Background
Complex numbers can be represented in the form of $ a + bi $ , where $ a $ is the real part and $ b $ is the imaginary part. $ i $ is representative of the value $ \sqrt{-1} $.
Complex numbers can also be represented in a Argand Plane with the coordinates $ (a, b) $. In Polar form, the complex number $ z $ can be written in the form $ z = r(\cos(x) + i \sin(x)) $ where x is the angle from the line $ a + bi $ and x axis.
Addition and Subtraction
For complex numbers, the addition and subtration only involves adding or subtracting the real part and the imaginary parts. For example :
$ ( a + bi) + (c + di) = (a + c) + (b + d) i $
Product
The product can be define such that the distributing law always hold :
$ ( a + bi) + (c + di) = a ( c + di) + bi ( c + di ) $
As $ i^2 = -1 $, thus
$ a ( c + di) + bi ( c + di ) = ac - bd + (ad + bc ) i $
Conjugates
Conjugates of complex numbers can be represented as the image of the original values on the real axis of an Argand plane. The following will hold for the conjugates of a complex number.
1. $ (a + b)' = a' + b' $
2. $ (ab)' = a'b' $
3. $ (z^n)' = (z')^n $
Thus the complex conjugates of $ z = a + bi $ is $ z' = a - bi $
We can perform division in complex numbers by multiplying the conjugates of the denominator to eliminate $ i $ in the denominator, simplifiying the calculation
Modulus
The absolute values of comples number equals to the squareroot of the sum of the squares of the imaginary and the real part.
$ |z| = \sqrt{a^2 + b^2} $
Thus, from the properties of complex numbers, we can also come to a conclusion that $ zz' = |z|^2 $
