8.3 Roots of Complex Numbers
Given a complex number z = r (cos q
+ i sin q) . All the roots of 'z' are
given by
where k = 0, 1, 2, 3, ...., (n1)
i) If k = 0 then
which is the principal nth root of z
ii) Also, for q = 0 and r = 1 then z = 1 and n^{th} root
of unity are given by
,k = 0, 1, 2, 3, ...., (n  1)
Example 1
Find three cube roots of unity.
Solution
n = 3 (1)^{1/3} =
Thus z^{3} = 1 and k = 0, 1, 2
when k = 0 z_{1} = cos (0) + i sin (0)
z_{1} = 1
when k = 1, z_{1} =
= cos (120^{0}) + sin (120^{0})
when k = 2 , z_{3 }
= cos (240^{0}) + i
sin (240^{0})
Hence the three cube roots of unity are 1,
Note that these roots are also denoted by 1, w and w2 respectively
Also, not that they are in geometric progression and we have
l + w + w^{2} = 0, w^{3} = 1 etc.
Example 2
Prove that the nth roots of unity form a G.P and each can be shown
as power of the other.
Solution
Let these roots be denoted by 1, w, w^{2}......, w^{n
1} forms a G. P. with common ratio
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