Question

Find the modulus and amplitude of the following complex number.

−4 − 4i

Answer 1

Let z = −4 − 4i.

Here, a = −4, b = −4 i.e., a < 0, b < 0

∴ |z| = `sqrt(a^2 + b^2)`

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

= `sqrt(16 + 16)`

= `sqrt(32)`

= `4sqrt(2)`

Here, (−4, −4) lies in 3^rd quadrant.

The amplitude (or argument) of a complex number z = a + bi is the angle θ formed with the positive real axis in the complex plane, and it is given by:

∴ θ = `tan^-1(b/a)`

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

= `tan^-1(1)`

= `tan^-1 (tan  pi/4)`

θ = `pi/4`

However, since the complex number −4−4i lies in the third quadrant (both real and imaginary parts are negative), the angle needs to be adjusted. The angle in the third quadrant is:

θ = `pi  + pi/4`

θ = `(5pi)/4`

Hence, modulus = `4sqrt2`  and

Amplitude (θ) = `(5pi)/4`