qth roots of a Complex number

One of the most important applications of De Moivre's theorem is to find the q^th roots of a complex number.

Let z = x + iy be a complex number.

Let z = r {cos q + i sin q} be its polar form.

We have z = r {cos(2np + q) + i sin (2np + q)}

[2np + q is the general amplitude]

where n = 0,1,2,3,........, q-1,....

Since we are finding the q^th roots of a complex number, z^1/q should satisfy the equation x^q = z which is an algebraic equation of degree q and should have exactly q roots.

where n = 0,1,2,3,........, q-1

By giving values for n = 0,1,2,3,........, q-1, we get q distinct q^th roots of z.

Cube roots of unity

where 2np + 0 is the general amplitude.

when n=0, z = cos 0 + i sin 0=1

\The cube roots of unity are which are usually denoted by 1, w, w².

Note 1:

Note 2:

Note 3:

Note 4:

Fourth roots of unity

Replacing the amplitude 0 by general amplitude 2np + 0, we get

when n = 0, z = cos0 + isin0 = 1

Note: