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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, ...., (n-1)

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 nth 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 z3 = 1 and k = 0, 1, 2

when k = 0 z1 = cos (0) + i sin (0)

z1 = 1

when k = 1, z1 =

= cos (1200) + sin (1200)

when k = 2 , z3

      = cos (2400) + i sin (2400)

            

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 + w2 = 0, w3 = 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, w2......, wn -1 forms a G. P. with common ratio

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Index

8.1 Geometry of complex numbers
8.2 De - Moivres's theorem
8.3 Roots of complex numbers
8.4 Cirsular functions of complex angles & hyperbolic function
Supplementary Problems

Chapter 9

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