QCE Specialist Maths: complex numbers in polar form

In polar form a complex number is r cis(θ), where r is its modulus and θ its argument. Multiplying multiplies the moduli and adds the arguments, so De Moivre's theorem follows: (r cis(θ))^n = r^n cis(nθ).

External assessment	One examination worth 50%, set and marked by the QCAA
Paper 1	Technology-free, 25%: short response and multiple choice
Paper 2	Technology-active, 25%: short response and multiple choice
Time	5 minutes perusal and 90 minutes working time for each paper; the formula book is provided
Internal assessment	Set by your school: a modelling task 20%, two short response examinations 15% each

The ideas that carry the topic

cis(θ) means cos(θ) + i sin(θ). The modulus of z = x + yi is the square root of x^2 + y^2, and its argument is the angle from the positive real axis, in the interval the question sets.
z times its conjugate is |z|^2. For a product, |z1 z2| = |z1| |z2| and arg(z1 z2) = arg(z1) + arg(z2), adjusted back into the required interval.
With De Moivre's theorem, (1 + i)^8 = (√2 cis(π/4))^8 = 16 cis(2π) = 16.
The n nth roots of a complex number share one modulus and are spaced 2π/n apart around a circle centred at the origin. For a polynomial with real coefficients, non-real roots come in conjugate pairs.

Where it is tested

Both papers can test any part of Units 3 and 4. Without a calculator, leave moduli and arguments exact, such as 2√2 and 3π/4; check the quadrant before you write an argument.

Try 3 questions

Paper 1 · Multiple choice · 1 mark · Further complex numbers
In polar form with the principal argument, $(\sqrt{3} + i)^{4}$ is
Answer it, every step marked
Paper 1 · Short response · 7 marks · Further complex numbers
Let $z = a + b i$ , where $a$ and $b$ are real and not both zero.
Answer it, every step marked
Paper 2 · Multiple choice · 1 mark · Further complex numbers
The fifth roots of unity are the vertices of a regular pentagon in the complex plane. Given that $\sin\left(\frac{2 \pi}{5}\right) = 0.9511$ , the area of the pentagon, correct to two decimal places, is
Answer it, every step marked

Next step

Take the free test Start the lesson

Questions people ask

What does cis mean?

cis(θ) is short for cos(θ) + i sin(θ). A complex number in polar form is r cis(θ), with modulus r and argument θ.

Which unit is this topic in?

Unit 3, Topic 1: Further complex numbers. Mathematical induction, which proves De Moivre's theorem, is Unit 3, Topic 2.

Can this topic appear in either paper?

Yes. Both papers can test any subject matter from Units 3 and 4.