February 29, 2020
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
An Imaginary number has a positive and a negative square root
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
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
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.
We'll find the cube roots of at an angle of
The 4th root of
The 5th Roots of -32 in Polar Form
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.