Roots of Unity (AQA A-Level Further Maths): Flashcards

$n$ complex numbers satisfying $z^n = 1$

z = e^{rac{2kpi i}{n}} where $k = 0, 1, ..., n-1$

1 (they lie on the unit circle)

$\frac{2\pi}{n}$ radians

0 (i.e., $1 + \omega + \omega^2 + ... + \omega^{n-1} = 0$)

$n$ solutions

Can be quoted without proof

$r^{\frac{1}{n}}e^{\frac{(\theta + 2k\pi)i}{n}}$, $k = 0, ..., n-1$

3 (one real, two complex)

$z = 1$