VCE Specialist Maths complex numbers

Write z in polar form, r cis(θ), and multiplying multiplies the moduli and adds the arguments. That one idea gives De Moivre's theorem for powers, the n equally spaced nth roots, and the geometry of the complex plane.

Examination 1	40 marks in 1 hour, no technology or notes
Examination 2	20 multiple-choice questions and 60 marks of extended answers, 2 hours
Study score	Examination 1 20%, Examination 2 40%, coursework 40%
Reading time	15 minutes before each examination; a formula sheet with both

The ideas that carry the topic

Polar form: the modulus |z| and the principal argument Arg(z), taken in the interval from minus pi (not included) to pi.
De Moivre's theorem: (r cis θ) to the power n is r to the n, cis nθ, for any integer n.
The nth roots of a complex number have the same modulus and arguments 2π/n apart, so they sit at the vertices of a regular n-sided polygon.
A polynomial with real coefficients has its non-real roots in conjugate pairs, so a known root gives a quadratic factor.

Try 3 questions

Examination 2 · Section A · 1 mark · Complex numbers in polar form
The principal argument of $-\sqrt{3} + i$ is
Answer it, every step marked
Examination 1 · No technology · 5 marks · Complex numbers in polar form
Let $z = 1 - \sqrt{3} i$ and $w = 2 i$ .
Answer it, every step marked
Examination 2 · Section B · 11 marks · Complex numbers in polar form
The tip of a robot arm moves in a horizontal plane, modelled as the complex plane with the pivot of the arm at the origin $O$ and distances in metres. The tip starts at $z_0 = 2 \sqrt{3} + 2 i$ . A turn command rotates the arm anticlockwise about $O$ through $\frac{\pi}{6}$ without changing its length, so after $n$ turn commands the tip is at $z_n = \left(\mathrm{cis}\left(\frac{\pi}{6}\right)\right)^{n} z_0$ .
Answer it, every step marked

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Questions people ask

Is the principal argument between 0 and 2π?

No. In VCE Specialist Mathematics the principal argument Arg(z) is in the interval from minus pi, not included, to pi, included.

Are complex numbers on both examinations?

They can be. Both examinations can test any content from Units 3 and 4.