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Geometry and arithmetic of complex numbers

Geometry and arithmetic of complex numbers

IB Year 6 | Grade 12
  1. Find the modulus and argument of the complex numbers \(p=\frac{1+i}{1-i}\) and \(q=\frac{\sqrt{2}}{1-i}\).
  2. On an Argand diagram with origin \(O\), sketch the points \(P\), \(Q\) and \(R\) representing the complex numbers \(p\), \(q\) and \(p+q\) respectively.
  3. Hence, prove that \(\arg\left(\frac{1+\sqrt{2}+i}{1-i}\right)=\frac{3\pi}{8}\).

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Answer:(a) \(|p|=|q|=1,\ \arg p=\frac\pi2,\ \arg q=\frac\pi4\). (b) \(P=(0,1),\ Q=\left(\frac1{\sqrt2},\frac1{\sqrt2}\right),\ R=\left(\frac1{\sqrt2},1+\frac1{\sqrt2}\right)\). (c) \(\frac{3\pi}8\).

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