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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 Index Chapter 9
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