Roots of complex numbers and roots of unityEdexcel A-Level Further Maths: Flashcards

What these 12 flashcards ask

State the formula for the nth roots of re^{i\theta}.
How many distinct nth roots does a non-zero complex number have?
What do the nth roots of a complex number form in the Argand diagram?
What is the modulus of every root of z^n=re^{i\theta}?
By what angle do consecutive roots differ?
State the nth roots of unity.
What is the sum of the nth roots of unity (n\ge2)?
If \omega=e^{2\pii/n}, what is \omega^n?
Express \omega^{n-k} in terms of \omega^k.
Cube roots of 8?
Side length of a regular n-gon with vertices on |z|=R?
Why must you write \theta+2k\pi before taking an nth root?

Exam questions on Roots of complex numbers and roots of unity

The complex number $w=8\mathrm{i}$ and the equation $z^3=8\mathrm{i}$ .
Find the three roots of $z^3=8\mathrm{i}$ , giving each in the form $x+\mathrm{i}y$ where $x$ and $y$ are exact.2 marks
Let $\omega=\mathrm{e}^{2\pi\mathrm{i}/5}$ , so that $1,\omega,\omega^2,\omega^3,\omega^4$ are the fifth roots of unity.
Show that $\omega^4=\overline{\omega}$ , and hence show that $\omega+\omega^4=2\cos\frac{2\pi}{5}$ .2 marks
The complex number $w=-4+4\sqrt3\,\mathrm{i}$ .
Write $w$ in the form $r\mathrm{e}^{\mathrm{i}\theta}$ , where $r>0$ and $-\pi<\theta\le\pi$ .3 marks

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).