Sum

Find the modulus and amplitude of the following complex numbers.

−4 − 4i

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#### Solution

Let z = −4 − 4i.

Here x = −4, y = −4

∴ |z| = `sqrt(x^2 + y^2)`

= `sqrt((-4)^2 + (-4)^2`

= `sqrt(16 + 16)`

= `sqrt(32)`

= `4sqrt(2)`

The amplitude θ is given by

θ = `tan^-1(y/x)`

= `tan^-1((-4)/(-4))`

= tan^{−1}(1)

= `tan^-1(tan pi/4)`

= `pi/4`

Hence, modulus = `4sqrt(2)` and

amplitude = tan^{−1}(1) or `(pi/4)`

Concept: Argand Diagram Or Complex Plane

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