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Modulus and Argument of a Complex Number

Modulus and Argument of a Complex Number

TGM Original Questions

Find the modulus and argument of each of the following complex numbers.

  1. \(1+\sqrt{3}\mathbf{i}\)
  2. \(4-3\mathbf{i}\)
  3. \(-\sqrt{3}+\mathbf{i}\)
  4. \(-1-\mathbf{i}\)
  5. \(-2\)
  6. \(-4\mathbf{i}\)

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Answer:\(\text{(i)}\ |z|=2,\ \arg z=\dfrac{\pi}{3}\) \(\text{(ii)}\ |z|=5,\ \arg z=-\tan^{-1}\!\left(\dfrac{3}{4}\right)\approx-0.644\) \(\text{(iii)}\ |z|=2,\ \arg z=\dfrac{5\pi}{6}\) \(\text{(iv)}\ |z|=\sqrt{2},\ \arg z=-\dfrac{3\pi}{4}\) \(\text{(v)}\ |z|=2,\ \arg z=\pi\) \(\text{(vi)}\ |z|=4,\ \arg z=-\dfrac{\pi}{2}\)

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