Chapters
Square Roots of Complex Numbers
Square Roots of Real and Imaginary Numbers
Just as we can find powers of a Complex number, we can also find any roots of a Complex number by using their Polar representation.
Taking the square root (or any root) of a Real number is the process of finding a Real number whose square is equal to the original number.
Real numbers have 2 square roots, a positive solution and its negative
Example
An Imaginary number has a positive and a negative square root
Example
Square Root of a Complex Number z=x+iy
When we want to find the square root of a Complex number, we are looking for a certain other Complex number which, when we square it, gives back the first Complex number as a result.
Complex numbers have 2 square roots, a certain Complex number 
and its opposite 
Existence of the Square Root
Example
This shows the existence of the square roots of a Complex number, but does not actually show the process we would use to solve for them.
Polar Form and the Power Formula
To do that, we need to use the Polar form of the Complex number and the power formula
and not use the restriction that 
be an Integer and open up its domain to include the Rational numbers 
If we choose 
and use the power formula, we get
Example
with 
and 
then
Square of the Opposite
If we square the opposite of the result 
, we should get back 
This shows that the 2 square roots are just opposites of each other 
and 
nth Root of a Complex Number
We can extend our result for the power to include 
by using the formula
where 
is one root out of the total for 
If 
then 
goes form 
to 
The solutions are all located the same distance from the origin and are all separated by the same angle, not necessarily as measured from the positive x-axis, but from the vector representing one root to the vector representing the next root in line.
Geometrically, each 
root of , , is an equally spaced point that forms the vertices of a regular polygon, i.e. each 
root of a Complex number forms a vertex of a pentagon.
Example
and 
then
Example
We'll find the cube roots of 
at an angle of 
with 
and 
Example
The 4th root of 
with 
and
The 5th Roots of -32 in Polar Form
with 
and 
The 8th Roots of Unity
The solutions to the equation 
are shown in the picture above: 
.
They are all spaced at an angle of 
apart as 
. If there were 6 roots, then the roots would be spread apart by an angle 
.
They all lie on the circumference of the Unit Circle a distance of 
unit away from the origin.
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this qwestion is necessary
l need support for complex numbers , inverse of quadratic equation, set theory