. . Finding n th roots of a complex number De Moivre's formula can be used to obtain roots of complex numbers. Suppose n is a positive integer and a complex number ω is n th root of z denoted by z / , then we have ω n = z .
...( ) Let ω = i sin z = r Since w is the n th root of z , then ω n = z ⇒ = r By De Moivre's theorem, = r Comparing the moduli and arguments, we get ρ n = r and n ρ = r / and k . Therefore, the values of ω are r / . Although there are infinitely many values of k , the distinct values of ω are obtained when , , , , . When k n n we get the same roots at regular intervals (cyclically).
Therefore the n th roots of complex number z cos are / / , , , , Complex Numbers If we set re , the formula for the n th roots of a complex number has a nice geometric interpretation, as shown in Figure. Note that because | the n roots all have the same modulus r they all lie on a circle of radius with centre at the origin. Furthermore, the n roots are equally spaced along the circle, because successive n roots have arguments that differ by p n . Remark ( ) General form of De Moivre's Theorem If x is rational, then cos is one of the values of (cos sin ) x .
( ) Polar form of unit circle Let z = e . Then, we get = cos ⇒ x iy = cos ⇒ x = . Therefore, z = represents a unit circle (radius one) centre at the origin.