Notation
$ a+bi $ where a and b are real numbers, and i is the imaginary unit, which has the property $ i^2 = -1 $. The real number a is called the real part of the complex number, and the real number b is the imaginary part.
Complex Arithmetic
Addition and Subtraction
$ (a + bi) + (c + di) = (a + c) + (b + d)i $
$ (a + bi) - (c + di) = (a - c) + (b - d)i $
Multiplication
$ (a + bi) * (c + di) = (ac - bd) + (ad + bc)i $
Division
$ \frac{(a + bi)}{(c + di)} = \left({ac + bd \over c^2 + d^2}\right) + \left( {bc - ad \over c^2 + d^2} \right)i\ $
The polar form
The polar form is $ z = r\,(\cos \varphi + i\sin \varphi )\, $.
It can also be represented as Euler's formula $ z = r\,\mathrm{e}^{i \varphi}\, $
