PolarformofaComplexnumber

# PolarformofaComplexnumber - Polar form of a Complex number...

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Polar form of a Complex number Example: Let z = 1 + i . Then Polar form of z = | z | e . For z = 1 + i, | z | = 1 2 + 1 2 = 2 and θ = π/ 4 . So z = 2 e i π 4 Multiplication: Let z 1 = | z 1 | e 1 ,z 2 = | z 2 | e 2 then z 1 z 2 = | z 1 || z 2 | e i ( θ 1 + θ 2 ) = | z 1 || z 2 | (cos ( θ 1 + θ 2 ) + i sin ( θ 1 + θ 2 )) Division: Let z 1 = | z 1 | e 1 ,z 2 = | z 2 | e 2 then z 1 z 2 = | z 1 | | z 2 | e i ( θ 1 - θ 2 ) = | z 1 | | z 2 | (cos ( θ 1 - θ 2 ) + i sin ( θ 1 - θ 2 )) Powers: Let z = | z | e then z n = | z | n e i ( ) = | z | n (cos ( ) + i sin ( )) Example: Let z=1+i, ﬁnd z 4 . Solution: We have z = 2 e i π 4 . Then z 4 = ( 2) 4 e = 4(cos ( π ) + i sin ( π )) = - 4 n-th roots of a complex number . Let z = | z | e . Then z 1 n = | z | 1 n e i ( θ +2 πk n ) = | z | 1 n (cos ( θ + 2 πk n ) + i sin ( θ + 2 πk n )) where k = 0 , 1 ,...,n
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