Question 3 · Complex numbers in polar and Euler form
Exact modulus and argument; product and quotient in polar form; De Moivre; conjugate roots
Maximum mark · 14 marksConsider the complex numbers $z_1 = 3 - 3i$ and $z_2 = -1 + \sqrt{3}\,i$.
- Write $z_1$ and $z_2$ in modulus–argument form, $r\operatorname{cis}\theta$, and in Euler form. Give the moduli and arguments exactly. 4 p
- Find $z_1z_2$ and $\dfrac{z_1}{z_2}$ in modulus–argument form. Also write $z_1z_2$ in Cartesian form, to three significant figures. 4 p
- Find $z_2^{6}$ in Cartesian form, justifying the method you use. 3 p
- Write down a quadratic equation with real coefficients that has $z_2$ as one of its roots. Justify your answer. 3 p
IB AI HL
TANS 1.12
TANS 1.13
Step-by-step solution
a) Polar and Euler form
$$|z_1| = \sqrt{3^2 + (-3)^2} = 3\sqrt{2}, \qquad \arg z_1 = -\dfrac{\pi}{4}\ \text{(fourth quadrant)}$$
$$|z_2| = \sqrt{(-1)^2 + (\sqrt{3})^2} = 2, \qquad \arg z_2 = \dfrac{2\pi}{3}\ \text{(second quadrant)}$$
$z_1 = 3\sqrt{2}\operatorname{cis}\left(-\frac{\pi}{4}\right) = 3\sqrt{2}\,e^{-\frac{\pi}{4}i}, \qquad z_2 = 2\operatorname{cis}\frac{2\pi}{3} = 2\,e^{\frac{2\pi}{3}i}$
Each modulus 1 mark, each argument 1 mark.
b) Product and quotient
Multiply (divide) the moduli and add (subtract) the arguments:
$$z_1z_2 = 6\sqrt{2}\operatorname{cis}\left(\dfrac{5\pi}{12}\right) \approx 2.20 + 8.20\,i, \qquad \dfrac{z_1}{z_2} = \dfrac{3\sqrt{2}}{2}\operatorname{cis}\left(-\dfrac{11\pi}{12}\right)$$
Product 2 marks, Cartesian form 1 mark, quotient 1 mark.
c) Sixth power
By De Moivre's theorem,
$$z_2^{6} = 2^{6}\operatorname{cis}\left(6\cdot\dfrac{2\pi}{3}\right) = 64\operatorname{cis}(4\pi) = 64$$
$z_2^6 = 64$
Use of De Moivre 1 mark, modulus 1 mark, argument and conclusion 1 mark.
d) Equation with real coefficients
With real coefficients, non-real roots come in conjugate pairs, so the other root is $\bar{z}_2 = -1 - \sqrt{3}\,i$. Their sum is $-2$ and their product is $|z_2|^2 = 4$, hence
$x^2 + 2x + 4 = 0$
Conjugate root 1 mark, sum and product 1 mark, equation 1 mark.