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प्रश्न
Find the modulus and amplitude of the following complex number.
−4 − 4i
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उत्तर
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 3rd 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`
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