is n = ( e e The product of all the n th roots of unity is n ( n . Since | , we have | ; hence ( ) n k ( ) , Therefore, n k Note ( ) All the n roots of n th roots unity are in Geometrical Progression ( ) Sum of the n roots of n th roots unity is always equal to zero. ( ) Product of the n roots of n th roots unity is equal to ( n . ( ) All the n roots of n th roots unity lie on the circumference of a circle whose centre is at the origin and radius equal to and these roots divide the circle into n equal parts and form a polygon of n sides.
Fig. . Re Im O - P Q m ω ω ω − ω n th roots of unity - - Complex Numbers Example . Find the cube roots of unity.
We have to find . Let z = then z = . In polar form, the equation z = can be written as e i k cos( sin( , , , . Therefore, z e i k + = , , Taking k = , , , we get, k = , z = cos k = , z = cos k = , z = cos Therefore, the cube roots of unity are −+ −− Þ , ω , and ω , where e i Example .
Find the fourth roots of unity. We have to find . Let z = . Then z = .
In polar form, the equation z = can be written as z = cos , , ,