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Question
Find the modulus and amplitude of the following complex numbers.
(1 + 2i)2 (1 − i)
Solution
Let z = (1 + 2i)2 (1 − i)
= (1 + 4i + 4i2) (1 − i)
= [1 + 4i + 4(−1)] (1 − i) ...[∵ i2 = −1]
= (−3 + 4i) (1 − i)
= −3 + 3i + 4i − 4i2
= −3 + 7i − 4 (−1)
= − 3 + 7i + 4
∴ z = 1 + 7i
∴ a = 1, b = 7, i. e. a > 0, b > 0
∴ |z| = `sqrt("a"^2 + "b"^2)`
= `sqrt(1^2 + 7^2)`
= `sqrt(1 + 49)`
= `5sqrt(2)`
Here, (1, 7) lies in 1st quadrant
amp (z) = `tan^-1("b"/"a")`
= tan–1(7)
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