Definition

Square root of the complex number z = a + bi, is called the complex number w = x + yi for which it is valid that or .

Example

Find the square root of z=5-12i.

Solution

The square root of z is the complex number x + yi for which 
We have:

So, or 2xy = 12 (1) or xy = 6 (2)
and 
 which is impossible or 
If x=3 the equation (2) gives y = 2
If x=-3 the equation (2) gives y = -2

Therefore the number z = 5 - 12i has two square roots: 3 + 2i, -3 - 2i
 

If z = a + bi a complex number and M(a,b) the corresponding element on the Complex plane.
If  and  then and .

Thus         Trigonometric form of z
The angle is called argument of z and is denoted arg(z).

Now, if , then easily and not using much trigonometry, it can be proved that:

Also the following can be proved:

De Moivre's Theorem

If n integer and a complex number, then 

For example:
.