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

What these 13 flashcards ask

How many distinct nth roots does a non-zero complex number have?
Formula for the nth roots of re^{i\theta}?
Where do the nth roots lie in the Argand diagram?
What shape do the nth roots form?
Angle between consecutive nth roots?
How do you get from one root to the next?
What are the nth roots of unity?
Sum of the nth roots of unity (n\ge2)?
Why is 1+\omega+\dots+\omega^{n-1}=0?
Complex conjugate of \omega^k?
Distance between adjacent vertices of the unit n-gon?
Product of distances from one vertex of a regular n-gon (circumradius R) to the others?
Area of a regular n-gon of circumradius R?

Exam questions on Roots of complex numbers and roots of unity

The equation $z^6=1$ .
Write down all six roots in the form $\mathrm{e}^{\mathrm{i}\theta}$ with $-\pi<\theta\le\pi$ .2 marks
The complex number $w=-8$ .
Find the three cube roots of $w$ in the form $a+b\mathrm{i}$ .2 marks
The equation $z^4=-8+8\sqrt3\,\mathrm{i}$ .
Find the modulus and the arguments, in the range $-\pi<\theta\le\pi$ , of the four roots of the equation.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).