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P.M./Complex/p.11
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(ii) ___________________________________________________________
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W.34 Q.6, 7
H.W. : W.34 Q.1 â 5
De Moivre's Theorem
If z = r(cos ï± + i sin ï± ) where r = z , for any rational number n, then
zn = r n(cos ï± + i sin ï± )n = r n(cos nï± + i sin nï± ) .
Proof :
Case 1 : Let n be a positive integer.
When n = 1, the statement is obviously true.
Assume zk = r k (cos kï± + i sin k ï±) .
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Case 2 : Let n be a positive rational number, i.e. n =
p and q. Let z
p
q
p
q
= r (cos ï¦ + i sin ï¦ )
p
for some positive relatively prime integers
q
where -ï° < ï¦ ï£ ï° .
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Case 3 : Let n be a negative rational number, then (-n) is a positive rational number. P.M./Complex/p.12
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