Here's how we add complex numbers:

(a₁ + ib₁) + (a₂ + ib₂) = (a₁ + a₂) + i(b₁ + b₂).

Hopefully, this reminds you of the parallelogram rule for adding vectors:

The absolute value of a+ib is

|a+ib| = √(a²+b²).

(Remember Pythagoras?) A useful definition is the complex conjugate

c* = a-ib

of c=a+ib. It makes it easy to calculate the absolute square |c|² of a complex number:

c c* = (a+ib) (a-ib) = a²+b² = |c|².

All you need to know in order to be able to multiply complex numbers is that i² = -1:

(a₁ + ib₁)(a₂ + ib₂) = (a₁a₂ - b₁b₂) + i(a₁b₂ + b₁a₂).

There is however an easier way of multiplying complex numbers (once you've gotten used to it). If α is the angle between the real axis and the straight line from the origin to c=a+ib, then

a = |c| cos(α) and b = |c| sin(α).

So we have that

c = |c| [cos(α) + i sin(α)].

The functioins sin(x) and cos(x) have the following expansions in powers of x:

sin(x) = x - (x³/3!) + (x⁵/5!) - (x⁷/7!) + ···,
cos(x) = 1 - (x²/2!) + (x⁴/4!) - (x⁶/6!) + ···.

(The factorial k! is defined (for 0 and the natural numbers) by 0!=1, 1!=1, 2!=1·2, 3!=1·2·3, and so on.) Hence

cos(x) + i sin(x) = 1 + ix - (x²/2!) - i(x³/3!) + (x⁴/4!) + i(x⁵/5!) - (x⁶/6!) - i(x⁷/7!)+ ···

This happens to be the expansion in powers of x of e^ix, where

e = e¹ = 1/0! + 1/1! + 1/2! + 1/3! + ··· = 2.7182818284590452353602874713526...

Thus

c = |c| e^iα.

The angle α is known as the phase of c, and e^iα is known as a phase factor. Observe that if |c|=1 then there is an α such that c = e^iα. In other words, the absolute value of a phase factor is 1.

Multiplying two complex numbers now boils down to multiplying their absolute values and adding their phases:

c₁ c₂ = |c₁|e^iα |c₂|e^iβ = |c₁| |c₂| e^i(α+β).