Complex Numbers


   
 
To find the qth roots of a Complex number
One of the most important applications of De Moivre's theorem is to find the qth 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 qth roots of a complex number, z1/q should satisfy the equation xq = 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 qth 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:
 
 
 
 
 
 
  
     
   
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