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Worksheet
Complex Numbers V: Roots of Unity
IB Mathematics AA · HL · Shadow Worksheet
Practice
Name
Question 1
Find all cube roots of unity and plot them on an Argand diagram.
Solve \(z^3=1\)
\(z=\text{cis}\dfrac{2k\pi}{3},\;k=0,1,2\)
Answer
\(z_0=1;\;z_1=\text{cis}\dfrac{2\pi}{3}=-\dfrac{1}{2}+\dfrac{\sqrt3}{2}i;\;z_2=\text{cis}\dfrac{4\pi}{3}=-\dfrac{1}{2}-\dfrac{\sqrt3}{2}i\)
Question 2
Let \(\omega=\text{cis}\dfrac{2\pi}{3}\). Show that \(1+\omega+\omega^2=0\).
Geometric series or direct
Sum of all 3rd roots of unity \(=\dfrac{\omega^3-1}{\omega-1}=\dfrac{0}{\omega-1}=0\) since \(\omega^3=1\) ✓
Question 3
Find all solutions to \(z^4=-16\), giving answers in polar and Cartesian form.
Write \(-16=16\,\text{cis}\,\pi\)
\(z=2\,\text{cis}\!\left(\dfrac{\pi+2k\pi}{4}\right),\;k=0,1,2,3\)
Four roots
\(k=0:\;2\,\text{cis}\tfrac{\pi}{4}=\sqrt2+\sqrt2 i\); \(k=1:\;2\,\text{cis}\tfrac{3\pi}{4}=-\sqrt2+\sqrt2 i\); \(k=2:\;2\,\text{cis}\tfrac{5\pi}{4}=-\sqrt2-\sqrt2 i\); \(k=3:\;2\,\text{cis}\tfrac{7\pi}{4}=\sqrt2-\sqrt2 i\)
Answer
\(\pm\sqrt{2}\pm\sqrt{2}\,i\) (all four sign combinations)
Question 4
Find all fifth roots of \(32i\) in polar form.
Write \(32i=32\,\text{cis}\tfrac{\pi}{2}\)
\(z=2\,\text{cis}\!\left(\dfrac{\pi/2+2k\pi}{5}\right)=2\,\text{cis}\!\left(\dfrac{\pi}{10}+\dfrac{2k\pi}{5}\right),\;k=0,1,2,3,4\)
Answer
\(2\,\text{cis}\dfrac{\pi}{10},\;2\,\text{cis}\dfrac{\pi}{2},\;2\,\text{cis}\dfrac{9\pi}{10},\;2\,\text{cis}\dfrac{13\pi}{10},\;2\,\text{cis}\dfrac{17\pi}{10}\)
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